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<title>Polynomial ring</title>

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Polynomial ring

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist" style="width: 20.5em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><a href="Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → Ring theory <a href="Ring_theory" title="Ring theory">Ring theory</a></th></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Basic concepts</div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="Ring_(mathematics)" title="Ring (mathematics)">Rings</a>

<dl><dd>• <a href="Subring" title="Subring">Subrings</a></dd>

<dd>• <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">Ideal</a></dd>

<dd>• <a href="Quotient_ring" title="Quotient ring">Quotient ring</a>

<dl><dd>• <a href="Fractional_ideal" title="Fractional ideal">Fractional ideal</a></dd>

<dd>• <a href="Total_ring_of_fractions" title="Total ring of fractions">Total ring of fractions</a></dd></dl></dd>

<dd>• <a href="Product_of_rings" title="Product of rings">Product of rings</a></dd>

<dd>• <a href="Free_product_of_associative_algebras" title="Free product of associative algebras">Free product of associative algebras</a></dd>

<dd>• <a href="Tensor_product_of_algebras" title="Tensor product of algebras">Tensor product of algebras</a></dd></dl>

<a href="Ring_homomorphism" title="Ring homomorphism">Ring homomorphisms</a>

<dl><dd>• <a href="Kernel_(algebra)#Ring_homomorphisms" title="Kernel (algebra)">Kernel</a></dd>

<dd>• <a href="Inner_automorphism#Ring_case" title="Inner automorphism">Inner automorphism</a></dd>

<dd>• <a href="Frobenius_endomorphism" title="Frobenius endomorphism">Frobenius endomorphism</a></dd></dl>

<a href="Algebraic_structure" title="Algebraic structure">Algebraic structures</a>

<dl><dd>• <a href="Module_(mathematics)" title="Module (mathematics)">Module</a></dd>

<dd>• <a href="Associative_algebra" title="Associative algebra">Associative algebra</a></dd>

<dd>• <a href="Graded_ring" title="Graded ring">Graded ring</a></dd>

<dd>• <a href="Involutive_ring" class="mw-redirect" title="Involutive ring">Involutive ring</a></dd>

<dd>• <a href="Category_of_rings" title="Category of rings">Category of rings</a>

<dl><dd>• <a href="Integer" title="Integer">Initial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

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<dd>• <a href="Zero_ring" title="Zero ring">Terminal ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }</annotation>

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Related structures

<dl><dd>• <a href="Field_(mathematics)" title="Field (mathematics)">Field</a>

<dl><dd>• <a href="Finite_field" title="Finite field">Finite field</a></dd></dl></dd>

<dd>• <a href="Non-associative_ring" class="mw-redirect" title="Non-associative ring">Non-associative ring</a>

<dd>• <a href="Jordan_ring" class="mw-redirect" title="Jordan ring">Jordan ring</a></dd></dl></dd>

<dd>• <a href="Semiring" title="Semiring">Semiring</a>

<dl><dd>• <a href="Semifield" title="Semifield">Semifield</a></dd></dl></dd></dl></div></div></td>

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<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="Commutative_algebra" title="Commutative algebra">Commutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="Commutative_ring" title="Commutative ring">Commutative rings</a>

<dl><dd>• <a href="Integral_domain" title="Integral domain">Integral domain</a>

<dl><dd>• <a href="Integrally_closed_domain" title="Integrally closed domain">Integrally closed domain</a></dd>

<dd>• <a href="GCD_domain" title="GCD domain">GCD domain</a></dd>

<dd>• <a href="Unique_factorization_domain" title="Unique factorization domain">Unique factorization domain</a></dd>

<dd>• <a href="Principal_ideal_domain" title="Principal ideal domain">Principal ideal domain</a></dd>

<dd>• <a href="Euclidean_domain" title="Euclidean domain">Euclidean domain</a></dd>

<dd>• <a href="Field_(mathematics)" title="Field (mathematics)">Field</a>

<dl><dd>• <a href="Finite_field" title="Finite field">Finite field</a></dd></dl></dd>

<dd>• <a href="Formal_power_series_ring" class="mw-redirect" title="Formal power series ring">Formal power series ring</a></dd></dl></dd></dl>

<a href="Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a>

<dl><dd>• <a href="Algebraic_number_field" title="Algebraic number field">Algebraic number field</a></dd>

<dd>• <a href="Integers_modulo_n" class="mw-redirect" title="Integers modulo n">Integers modulo n</a></dd>

<dd>• <a href="Ring_of_integers" title="Ring of integers">Ring of integers</a></dd>

<dd>• <a href="P-adic_integer" class="mw-redirect" title="P-adic integer">p-adic integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}">

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<dd>• <a href="P-adic_number" title="P-adic number">p-adic numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}">

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<dd>• <a href="Pr%C3%BCfer_group#The_Prüfer_group_as_a_ring" title="Prüfer group">Prüfer p-ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} (p^{\infty })}">

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="Noncommutative_algebra" class="mw-redirect" title="Noncommutative algebra">Noncommutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="Noncommutative_ring" title="Noncommutative ring">Noncommutative rings</a>

<dl><dd>• <a href="Division_ring" title="Division ring">Division ring</a></dd>

<dd>• <a href="Semiprimitive_ring" title="Semiprimitive ring">Semiprimitive ring</a></dd>

<dd>• <a href="Simple_ring" title="Simple ring">Simple ring</a></dd>

<dd>• <a href="Commutator_(ring_theory)" class="mw-redirect" title="Commutator (ring theory)">Commutator</a></dd></dl>

<a href="Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a>

<a href="Free_algebra" title="Free algebra">Free algebra</a>

<a href="Clifford_algebra" title="Clifford algebra">Clifford algebra</a>

<dl><dd>• <a href="Geometric_algebra" title="Geometric algebra">Geometric algebra</a></dd></dl>

<a href="Operator_algebra" title="Operator algebra">Operator algebra</a></div></div></td>

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In <a href="Mathematics" title="Mathematics">mathematics</a>, especially in the field of <a href="Algebra" title="Algebra">algebra</a>, a polynomial ring or polynomial algebra is a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> formed from the <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of <a href="Polynomial" title="Polynomial">polynomials</a> in one or more indeterminates (traditionally also called <a href="Variable_(mathematics)" title="Variable (mathematics)">variables</a>) with <a href="Coefficient" title="Coefficient">coefficients</a> in another <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>, often a <a href="Field_(mathematics)" title="Field (mathematics)">field</a>.

Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the <a href="Integer#Algebraic_properties" title="Integer">integers</a>.

Polynomial rings occur and are often fundamental in many parts of mathematics such as <a href="Number_theory" title="Number theory">number theory</a>, <a href="Commutative_algebra" title="Commutative algebra">commutative algebra</a>, and <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. In <a href="Ring_theory" title="Ring theory">ring theory</a>, many classes of rings, such as <a href="Unique_factorization_domain" title="Unique factorization domain">unique factorization domains</a>, <a href="Regular_ring" class="mw-redirect" title="Regular ring">regular rings</a>, <a href="Group_ring" title="Group ring">group rings</a>, <a href="Formal_power_series" title="Formal power series">rings of formal power series</a>, <a href="Ore_polynomial" class="mw-redirect" title="Ore polynomial">Ore polynomials</a>, <a href="Graded_ring" title="Graded ring">graded rings</a>, have been introduced for generalizing some properties of polynomial rings.

A closely related notion is that of the <a href="Ring_of_polynomial_functions" title="Ring of polynomial functions">ring of polynomial functions</a> on a <a href="Vector_space" title="Vector space">vector space</a>, and, more generally, <a href="Ring_of_regular_functions" class="mw-redirect" title="Ring of regular functions">ring of regular functions</a> on an <a href="Algebraic_variety" title="Algebraic variety">algebraic variety</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition_(univariate_case)">Definition (univariate case)</h2></div>

Let K be a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> or (more generally) a <a href="Commutative_ring" title="Commutative ring">commutative ring</a>.

The polynomial ring in X over K, which is denoted K[X], can be defined in several equivalent ways. One of them is to define K[X] as the set of expressions, called polynomials in X, of the form<a href="#cite_note-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m-1}X^{m-1}+p_{m}X^{m},}</annotation>

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where m is a nonnegative integer, the coefficients p0, p1, …, pm of p are elements of K, pm ≠ 0 if m > 0, and X, X2, …, are symbols called "powers" of X that follow the usual rules of <a href="Exponent" class="mw-redirect" title="Exponent">exponents</a>: X0 = 1, X1 = X, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{k}\cdot X^{l}=X^{k+l}}">

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<annotation encoding="application/x-tex">{\displaystyle X^{k}\cdot X^{l}=X^{k+l}}</annotation>

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Two polynomials are equal when the corresponding coefficients of each Xk are equal.

One can think of the ring K[X] as arising from K by adding one new element X that is external to K, commutes with all elements of K, and has no other specific properties. This can be used for an equivalent definition of polynomial rings.

The polynomial ring in X over K is equipped with an addition, a multiplication and a <a href="Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a> that make it a <a href="Commutative_algebra_(structure)" class="mw-redirect" title="Commutative algebra (structure)">commutative algebra</a>. These operations are defined according to the ordinary rules for manipulating algebraic expressions. Specifically, if

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<annotation encoding="application/x-tex">{\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m}X^{m},}</annotation>

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and

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<annotation encoding="application/x-tex">{\displaystyle q=q_{0}+q_{1}X+q_{2}X^{2}+\cdots +q_{n}X^{n},}</annotation>

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</math><img src="./73f0ed4fbf183ca81f8f41d979ce6898f60e594c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.675ex; height:3.009ex;" alt="{\displaystyle q=q_{0}+q_{1}X+q_{2}X^{2}+\cdots +q_{n}X^{n},}" loading="lazy"></dd></dl>

then

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<annotation encoding="application/x-tex">{\displaystyle p+q=r_{0}+r_{1}X+r_{2}X^{2}+\cdots +r_{k}X^{k},}</annotation>

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</math><img src="./212799bcd3c47b0685ce5c9509d302d33838d3e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:39.562ex; height:3.009ex;" alt="{\displaystyle p+q=r_{0}+r_{1}X+r_{2}X^{2}+\cdots +r_{k}X^{k},}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle pq=s_{0}+s_{1}X+s_{2}X^{2}+\cdots +s_{l}X^{l},}</annotation>

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</math><img src="./43415c6a0f0b41cd58ab70d94ba9d4bba6ed4dcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:36.156ex; height:3.009ex;" alt="{\displaystyle pq=s_{0}+s_{1}X+s_{2}X^{2}+\cdots +s_{l}X^{l},}" loading="lazy"></dd></dl>

where k = max(m, n), l = m + n,

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<annotation encoding="application/x-tex">{\displaystyle r_{i}=p_{i}+q_{i}}</annotation>

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</math><img src="./820e0847b3d24cf28bd9cd6fd7432dbe1a71c009.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.593ex; height:2.343ex;" alt="{\displaystyle r_{i}=p_{i}+q_{i}}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle s_{i}=p_{0}q_{i}+p_{1}q_{i-1}+\cdots +p_{i}q_{0}.}</annotation>

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</math><img src="./6ad837fcb4f1dead70c7ab66c140a86cc6b1fc9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.162ex; height:2.343ex;" alt="{\displaystyle s_{i}=p_{0}q_{i}+p_{1}q_{i-1}+\cdots +p_{i}q_{0}.}" loading="lazy"></dd></dl>

In these formulas, the polynomials p and q are extended by adding "dummy terms" with zero coefficients, so that all pi and qi that appear in the formulas are defined. Specifically, if m < n, then pi = 0 for m < i ≤ n.

The scalar multiplication is the special case of the multiplication where p = p0 is reduced to its constant term (the term that is independent of X); that is

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<annotation encoding="application/x-tex">{\displaystyle p_{0}\left(q_{0}+q_{1}X+\dots +q_{n}X^{n}\right)=p_{0}q_{0}+\left(p_{0}q_{1}\right)X+\cdots +\left(p_{0}q_{n}\right)X^{n}}</annotation>

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</math><img src="./61f24f40ae70799b47b9bc9d23a862398813af44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:64.427ex; height:2.843ex;" alt="{\displaystyle p_{0}\left(q_{0}+q_{1}X+\dots +q_{n}X^{n}\right)=p_{0}q_{0}+\left(p_{0}q_{1}\right)X+\cdots +\left(p_{0}q_{n}\right)X^{n}}" loading="lazy"></dd></dl>

It is straightforward to verify that these three operations satisfy the axioms of a commutative algebra over K. Therefore, polynomial rings are also called polynomial algebras.

Another equivalent definition is often preferred, although less intuitive, because it is easier to make it completely rigorous, which consists in defining a polynomial as an infinite <a href="Sequence" title="Sequence">sequence</a> (p0, p1, p2, …) of elements of K, having the property that only a finite number of the elements are nonzero, or equivalently, a sequence for which there is some m so that pn = 0 for n > m. In this case, p0 and X are considered as alternate notations for

the sequences (p0, 0, 0, …) and (0, 1, 0, 0, …), respectively. A straightforward use of the operation rules shows that the expression

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<annotation encoding="application/x-tex">{\displaystyle p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m}X^{m}}</annotation>

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</math><img src="./5b97e40e088d4e02ffe6dafdaa9d1bbc28c067fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:32.393ex; height:3.009ex;" alt="{\displaystyle p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m}X^{m}}" loading="lazy"></dd></dl>

is then an alternate notation for the sequence

<div class="mw-heading mw-heading3"><h3 id="Terminology">Terminology</h3></div>

Let

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<annotation encoding="application/x-tex">{\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m-1}X^{m-1}+p_{m}X^{m},}</annotation>

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</math><img src="./42a98a1580574bb492732b0c7b39ea8654b64c0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:50.865ex; height:3.009ex;" alt="{\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m-1}X^{m-1}+p_{m}X^{m},}" loading="lazy"></dd></dl>

be a nonzero polynomial with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{m}\neq 0}">

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</math><img src="./a00dacba9af9ae12355de9d20799276cdc5e9a94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:7.195ex; height:2.676ex;" alt="{\displaystyle p_{m}\neq 0}" loading="lazy">

The constant term of p is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0}.}">

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<annotation encoding="application/x-tex">{\displaystyle p_{0}.}</annotation>

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</math><img src="./fc6a990c974dd2af4bc7a4f28165f8e151895eec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.96ex; height:2.009ex;" alt="{\displaystyle p_{0}.}" loading="lazy"> It is zero in the case of the zero polynomial.

The degree of p, written deg(p) is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,}">

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<annotation encoding="application/x-tex">{\displaystyle m,}</annotation>

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</math><img src="./dad66d19bb37bc69223cb004be2ea5dd95f9564c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.687ex; height:2.009ex;" alt="{\displaystyle m,}" loading="lazy"> the largest k such that the coefficient of Xk is not zero.<a href="#cite_note-4">[4]</a>

The leading coefficient of p is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{m}.}">

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<annotation encoding="application/x-tex">{\displaystyle p_{m}.}</annotation>

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</math><img src="./38525d102a41271ff9ea650117ab3185bbc99d91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.581ex; height:2.009ex;" alt="{\displaystyle p_{m}.}" loading="lazy"><a href="#cite_note-5">[5]</a>

In the special case of the zero polynomial, all of whose coefficients are zero, the leading coefficient is undefined, and the degree has been variously left undefined,<a href="#cite_note-6">[6]</a> defined to be −1,<a href="#cite_note-7">[7]</a> or defined to be a −∞.<a href="#cite_note-8">[8]</a>

A constant polynomial is either the zero polynomial, or a polynomial of degree zero.

A nonzero polynomial is <a href="Monic_polynomial" title="Monic polynomial">monic</a> if its leading coefficient is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1.}">

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</math><img src="./af8c4e445819b13a052647aa3eb2be990b0a4b24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 1.}" loading="lazy">

Given two polynomials p and q, if the degree of the zero polynomial is defined to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty ,}">

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<annotation encoding="application/x-tex">{\displaystyle -\infty ,}</annotation>

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</math><img src="./79d5ae78431259d0588140f827d0a1fcdfa4ec7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.779ex; height:2.343ex;" alt="{\displaystyle -\infty ,}" loading="lazy"> one has

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<annotation encoding="application/x-tex">{\displaystyle \deg(p+q)\leq \max(\deg(p),\deg(q)),}</annotation>

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</math><img src="./d799c7bc1b1d5b6ab82844fd7ff229e92fcafa01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.123ex; height:2.843ex;" alt="{\displaystyle \deg(p+q)\leq \max(\deg(p),\deg(q)),}" loading="lazy"></dd></dl>

and, over a <a href="Field_(mathematics)" title="Field (mathematics)">field</a>, or more generally an <a href="Integral_domain" title="Integral domain">integral domain</a>,<a href="#cite_note-9">[9]</a>

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<annotation encoding="application/x-tex">{\displaystyle \deg(pq)=\deg(p)+\deg(q).}</annotation>

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</math><img src="./fda60db0a9829e4489a7f2951746dd377e2942d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.954ex; height:2.843ex;" alt="{\displaystyle \deg(pq)=\deg(p)+\deg(q).}" loading="lazy"></dd></dl>

It follows immediately that, if K is an integral domain, then so is K[X].<a href="#cite_note-10">[10]</a>

It follows also that, if K is an integral domain, a polynomial is a <a href="Unit_(ring_theory)" title="Unit (ring theory)">unit</a> (that is, it has a <a href="Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a>) if and only if it is constant and is a unit in K.

Two polynomials are <a href="Associated_element" class="mw-redirect" title="Associated element">associated</a> if either one is the product of the other by a unit.

Over a field, every nonzero polynomial is associated to a unique monic polynomial.

Given two polynomials, p and q, one says that p divides q, p is a divisor of q, or q is a multiple of p, if there is a polynomial r such that q = pr.

A polynomial is <a href="Irreducible_polynomial" title="Irreducible polynomial">irreducible</a> if it is not the product of two non-constant polynomials, or equivalently, if its divisors are either constant polynomials or have the same degree.

<div class="mw-heading mw-heading3"><h3 id="Polynomial_evaluation">Polynomial evaluation</h3></div>

/* start https://en.wikipedia.org/ */

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</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Polynomial_evaluation" title="Polynomial evaluation">Polynomial evaluation</a></div>

Let K be a field or, more generally, a <a href="Commutative_ring" title="Commutative ring">commutative ring</a>, and R a ring containing K. For any polynomial P in K[X] and any element a in R, the substitution of X with a in P defines an element of R, which is <a href="Polynomial_notation" class="mw-redirect" title="Polynomial notation">denoted</a> P(a). This element is obtained by carrying on in R after the substitution the operations indicated by the expression of the polynomial. This computation is called the evaluation of P at a. For example, if we have

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<annotation encoding="application/x-tex">{\displaystyle P=X^{2}-1,}</annotation>

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</math><img src="./d72ad31270fc7a6ba44669739a801a13b4edaff9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.545ex; height:3.009ex;" alt="{\displaystyle P=X^{2}-1,}" loading="lazy"></dd></dl>

we have

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P(3)&=3^{2}-1=8,\\P(X^{2}+1)&=\left(X^{2}+1\right)^{2}-1=X^{4}+2X^{2}\end{aligned}}}</annotation>

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</math><img src="./21ded8d535505a93146c397bad1cfdee8888c50b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.751ex; margin-bottom: -0.253ex; width:41.903ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}P(3)&=3^{2}-1=8,\\P(X^{2}+1)&=\left(X^{2}+1\right)^{2}-1=X^{4}+2X^{2}\end{aligned}}}" loading="lazy"></dd></dl>

(in the first example R = K, and in the second one R = K[X]). Substituting X for itself results in

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</math><img src="./daf0dded030cff79ca51db34ecdde372f3b5d283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.025ex; height:2.843ex;" alt="{\displaystyle P=P(X),}" loading="lazy"></dd></dl>

explaining why the sentences "Let P be a polynomial" and "Let P(X) be a polynomial" are equivalent.

The polynomial function defined by a polynomial P is the function from K into K that is defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto P(x).}">

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<annotation encoding="application/x-tex">{\displaystyle x\mapsto P(x).}</annotation>

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</math><img src="./cf63b200afc3d9b7cb993fbc51c8cf051f957f0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.475ex; height:2.843ex;" alt="{\displaystyle x\mapsto P(x).}" loading="lazy"> If K is an infinite field, two different polynomials define different polynomial functions, but this property is false for finite fields. For example, if K is a field with q elements, then the polynomials 0 and Xq − X both define the zero function.

For every a in R, the evaluation at a, that is, the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\mapsto P(a)}">

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</math><img src="./a75ca1466229080cda9212a275e518277ee2a49b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.144ex; height:2.843ex;" alt="{\displaystyle P\mapsto P(a)}" loading="lazy"> defines an <a href="Algebra_homomorphism" class="mw-redirect" title="Algebra homomorphism">algebra homomorphism</a> from K[X] to R, which is the unique homomorphism from K[X] to R that fixes K, and maps X to a. In other words, K[X] has the following <a href="Universal_property" title="Universal property">universal property</a>:

<dl><dd>For every ring R containing K, and every element a of R, there is a unique algebra homomorphism from K[X] to R that fixes K, and maps X to a.</dd></dl>

As for all universal properties, this defines the pair (K[X], X) up to a unique isomorphism, and can therefore be taken as a definition of K[X].

The <a href="Image_(mathematics)" title="Image (mathematics)">image</a> of the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\mapsto P(a)}">

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</math><img src="./a75ca1466229080cda9212a275e518277ee2a49b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.144ex; height:2.843ex;" alt="{\displaystyle P\mapsto P(a)}" loading="lazy">, that is, the subset of R obtained by substituting a for X in elements of K[X], is denoted K[a].<a href="#cite_note-11">[11]</a> For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [{\sqrt {2}}]=\{P({\sqrt {2}})\mid P(X)\in \mathbb {Z} [X]\}}">

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</math><img src="./aae2cec3ca445fa760df51ba17d48477fcb6c729.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.155ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} [{\sqrt {2}}]=\{P({\sqrt {2}})\mid P(X)\in \mathbb {Z} [X]\}}" loading="lazy">, and the simplification rules for the powers of a square root imply <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [{\sqrt {2}}]=\{a+b{\sqrt {2}}\mid a\in \mathbb {Z} ,b\in \mathbb {Z} \}.}">

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<div class="mw-heading mw-heading2"><h2 id="Univariate_polynomials_over_a_field">Univariate polynomials over a field</h2></div>

If K is a <a href="Field_(mathematics)" title="Field (mathematics)">field</a>, the polynomial ring K[X] has many properties that are similar to those of the <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> of integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} .}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} .}</annotation>

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</math><img src="./89f4f38f32c2068bca9dc701d13b03dd4a5d52ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} .}" loading="lazy"> Most of these similarities result from the similarity between the <a href="Long_division" title="Long division">long division of integers</a> and the <a href="Polynomial_long_division" title="Polynomial long division">long division of polynomials</a>.

Most of the properties of K[X] that are listed in this section do not remain true if K is not a field, or if one considers polynomials in several indeterminates.

Like for integers, the <a href="Euclidean_division_of_polynomials" class="mw-redirect" title="Euclidean division of polynomials">Euclidean division of polynomials</a> has a property of uniqueness. That is, given two polynomials a and b ≠ 0 in K[X], there is a unique pair (q, r) of polynomials such that a = bq + r, and either r = 0 or deg(r) < deg(b). This makes K[X] a <a href="Euclidean_domain" title="Euclidean domain">Euclidean domain</a>. However, most other Euclidean domains (except integers) do not have any property of uniqueness for the division nor an easy algorithm (such as long division) for computing the Euclidean division.

The Euclidean division is the basis of the <a href="Euclidean_algorithm_for_polynomials" class="mw-redirect" title="Euclidean algorithm for polynomials">Euclidean algorithm for polynomials</a> that computes a <a href="Polynomial_greatest_common_divisor" title="Polynomial greatest common divisor">polynomial greatest common divisor</a> of two polynomials. Here, "greatest" means "having a maximal degree" or, equivalently, being maximal for the <a href="Preorder" title="Preorder">preorder</a> defined by the degree. Given a greatest common divisor of two polynomials, the other greatest common divisors are obtained by multiplication by a nonzero constant (that is, all greatest common divisors of a and b are associated). In particular, two polynomials that are not both zero have a unique greatest common divisor that is monic (leading coefficient equal to 1).

The <a href="Extended_Euclidean_algorithm" title="Extended Euclidean algorithm">extended Euclidean algorithm</a> allows computing (and proving) <a href="B%C3%A9zout's_identity" title="Bézout's identity">Bézout's identity</a>. In the case of K[X], it may be stated as follows. Given two polynomials p and q of respective degrees m and n, if their monic greatest common divisor g has the degree d, then there is a unique pair (a, b) of polynomials such that

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<annotation encoding="application/x-tex">{\displaystyle ap+bq=g,}</annotation>

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</math><img src="./f92a84e9630875a9835660aaf79cd12489630fcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.168ex; height:2.509ex;" alt="{\displaystyle ap+bq=g,}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle \deg(a)\leq n-d,\quad \deg(b)<m-d.}</annotation>

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</math><img src="./3fcf14ca3771d0fca89bf73f367c084213b8acdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.569ex; height:2.843ex;" alt="{\displaystyle \deg(a)\leq n-d,\quad \deg(b)<m-d.}" loading="lazy"></dd></dl>

(For making this true in the limiting case where m = d or n = d, one has to define as negative the degree of the zero polynomial. Moreover, the equality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(a)=n-d}">

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<annotation encoding="application/x-tex">{\displaystyle \deg(a)=n-d}</annotation>

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</math><img src="./3e7f803022b2b0220c8d7d9351563ddbfb49a1e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.076ex; height:2.843ex;" alt="{\displaystyle \deg(a)=n-d}" loading="lazy"> can occur only if p and q are associated.) The uniqueness property is rather specific to K[X]. In the case of the integers the same property is true, if degrees are replaced by absolute values, but, for having uniqueness, one must require a > 0.

<a href="Euclid's_lemma" title="Euclid's lemma">Euclid's lemma</a> applies to K[X]. That is, if a divides bc, and is <a href="Coprime" class="mw-redirect" title="Coprime">coprime</a> with b, then a divides c. Here, coprime means that the monic greatest common divisor is 1. Proof: By hypothesis and Bézout's identity, there are e, p, and q such that ae = bc and 1 = ap + bq. So

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<annotation encoding="application/x-tex">{\displaystyle c=c(ap+bq)=cap+aeq=a(cp+eq).}</annotation>

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</math><img src="./31a6c85cd218f44e9a59a4fc3f6833193d63a8cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.91ex; height:2.843ex;" alt="{\displaystyle c=c(ap+bq)=cap+aeq=a(cp+eq).}" loading="lazy">

The <a href="Unique_factorization" class="mw-redirect" title="Unique factorization">unique factorization</a> property results from Euclid's lemma. In the case of integers, this is the <a href="Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>. In the case of K[X], it may be stated as: every non-constant polynomial can be expressed in a unique way as the product of a constant, and one or several irreducible monic polynomials; this decomposition is unique up to the order of the factors. In other terms K[X] is a <a href="Unique_factorization_domain" title="Unique factorization domain">unique factorization domain</a>. If K is the field of complex numbers, the <a href="Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a> asserts that a univariate polynomial is irreducible if and only if its degree is one. In this case the unique factorization property can be restated as: every non-constant univariate polynomial over the complex numbers can be expressed in a unique way as the product of a constant, and one or several polynomials of the form X − r; this decomposition is unique up to the order of the factors. For each factor, r is a <a href="Root_of_a_function" class="mw-redirect" title="Root of a function">root</a> of the polynomial, and the number of occurrences of a factor is the <a href="Multiplicity_(mathematics)" title="Multiplicity (mathematics)">multiplicity</a> of the corresponding root.

<div class="mw-heading mw-heading3"><h3 id="Derivation">Derivation</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Formal_derivative" title="Formal derivative">Formal derivative</a> and <a href="Derivation_(differential_algebra)" title="Derivation (differential algebra)">Derivation (differential algebra)</a></div>

The <a href="Formal_derivative" title="Formal derivative">(formal) derivative</a> of the polynomial

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<annotation encoding="application/x-tex">{\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}}</annotation>

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</math><img src="./11d5ad0928f318b0c3fa0681a9ec7d302d48b72a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.632ex; height:3.009ex;" alt="{\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}}" loading="lazy"></dd></dl>

is the polynomial

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<annotation encoding="application/x-tex">{\displaystyle a_{1}+2a_{2}X+\cdots +na_{n}X^{n-1}.}</annotation>

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</math><img src="./cc369d74d6b6261ed4157aa8d868885495ca42c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.761ex; height:3.009ex;" alt="{\displaystyle a_{1}+2a_{2}X+\cdots +na_{n}X^{n-1}.}" loading="lazy"></dd></dl>

In the case of polynomials with <a href="Real_number" title="Real number">real</a> or <a href="Complex_number" title="Complex number">complex</a> coefficients, this is the standard <a href="Derivative" title="Derivative">derivative</a>. The above formula defines the derivative of a polynomial even if the coefficients belong to a ring on which no notion of <a href="Limit_(mathematics)" title="Limit (mathematics)">limit</a> is defined. The derivative makes the polynomial ring a <a href="Differential_algebra" title="Differential algebra">differential algebra</a>.

The existence of the derivative is one of the main properties of a polynomial ring that is not shared with integers, and makes some computations easier on a polynomial ring than on integers.

<div class="mw-heading mw-heading4"><h4 id="Square-free_factorization">Square-free factorization</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Square-free_polynomial" title="Square-free polynomial">Square-free polynomial</a></div>

A polynomial with coefficients in a field or integral domain is square-free if it does not have a <a href="Multiple_root" class="mw-redirect" title="Multiple root">multiple root</a> in the <a href="Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a> containing its coefficients. In particular, a polynomial of degree n with real or complex coefficients is square-free if it has n distinct complex roots. Equivalently, a polynomial over a field is square-free if and only if the <a href="Polynomial_greatest_common_divisor" title="Polynomial greatest common divisor">greatest common divisor</a> of the polynomial and its derivative is 1.

A square-free factorization of a polynomial is an expression for that polynomial as a product of powers of <a href="Pairwise_relatively_prime" class="mw-redirect" title="Pairwise relatively prime">pairwise relatively prime</a> square-free factors. Over the real numbers (or any other field of <a href="Characteristic_0" class="mw-redirect" title="Characteristic 0">characteristic 0</a>), such a factorization can be computed efficiently by <a href="Yun's_algorithm" class="mw-redirect" title="Yun's algorithm">Yun's algorithm</a>. Less efficient algorithms are known for <a href="Factorization_of_polynomials_over_finite_fields#Square-free_factorization" title="Factorization of polynomials over finite fields">square-free factorization of polynomials over finite fields</a>.

<div class="mw-heading mw-heading4"><h4 id="Lagrange_interpolation">Lagrange interpolation</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Lagrange_polynomial" title="Lagrange polynomial">Lagrange polynomial</a></div>

Given a finite set of ordered pairs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{j},y_{j})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{j},y_{j})}</annotation>

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</math><img src="./dec8853489ad3dc2385adf0a049a75aed4dc0987.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.132ex; height:3.009ex;" alt="{\displaystyle (x_{j},y_{j})}" loading="lazy"> with entries in a field and distinct values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation>

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</math><img src="./5db47cb3d2f9496205a17a6856c91c1d3d363ccd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}" loading="lazy">, among the polynomials <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

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</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> that interpolate these points (so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{j})=y_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle f(x_{j})=y_{j}}</annotation>

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</math><img src="./19bff0c1d3e42fb1578c06cd96386a392256f2b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.475ex; height:3.009ex;" alt="{\displaystyle f(x_{j})=y_{j}}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

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<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

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</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">), there is a unique polynomial of smallest degree. This is the Lagrange interpolation polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(x)}">

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<annotation encoding="application/x-tex">{\displaystyle L(x)}</annotation>

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</math><img src="./88b78cb2b3a60d964f3016e94f0d7bd081ebd3be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.722ex; height:2.843ex;" alt="{\displaystyle L(x)}" loading="lazy">. If there are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> ordered pairs, the degree of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(x)}">

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<annotation encoding="application/x-tex">{\displaystyle L(x)}</annotation>

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</math><img src="./88b78cb2b3a60d964f3016e94f0d7bd081ebd3be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.722ex; height:2.843ex;" alt="{\displaystyle L(x)}" loading="lazy"> is at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k-1}">

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<annotation encoding="application/x-tex">{\displaystyle k-1}</annotation>

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</math><img src="./21363ebd7038c93aae93127e7d910fc1b2e2c745.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k-1}" loading="lazy">. The polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(x)}">

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<annotation encoding="application/x-tex">{\displaystyle L(x)}</annotation>

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</math><img src="./88b78cb2b3a60d964f3016e94f0d7bd081ebd3be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.722ex; height:2.843ex;" alt="{\displaystyle L(x)}" loading="lazy"> can be computed explicitly in terms of the input data <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{j},y_{j})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{j},y_{j})}</annotation>

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</math><img src="./dec8853489ad3dc2385adf0a049a75aed4dc0987.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.132ex; height:3.009ex;" alt="{\displaystyle (x_{j},y_{j})}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Polynomial_decomposition">Polynomial decomposition</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Polynomial_decomposition" title="Polynomial decomposition">Polynomial decomposition</a></div>

A decomposition of a polynomial is a way of expressing it as a <a href="Function_composition" title="Function composition">composition</a> of other polynomials of degree larger than 1. A polynomial that cannot be decomposed is indecomposable. <a href="Ritt's_polynomial_decomposition_theorem" class="mw-redirect" title="Ritt's polynomial decomposition theorem">Ritt's polynomial decomposition theorem</a> asserts that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=g_{1}\circ g_{2}\circ \cdots \circ g_{m}=h_{1}\circ h_{2}\circ \cdots \circ h_{n}}">

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</math><img src="./3a1b9544c4e2e2d676a26b27b528101791b7faa8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:40.545ex; height:2.509ex;" alt="{\displaystyle f=g_{1}\circ g_{2}\circ \cdots \circ g_{m}=h_{1}\circ h_{2}\circ \cdots \circ h_{n}}" loading="lazy"> are two different decompositions of the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=n}">

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</math><img src="./69c9d8e54796e7de7d4738510cc10bc3fc55d48e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.676ex;" alt="{\displaystyle m=n}" loading="lazy"> and the degrees of the indecomposables in one decomposition are the same as the degrees of the indecomposables in the other decomposition (though not necessarily in the same order).

<div class="mw-heading mw-heading3"><h3 id="Factorization">Factorization</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Polynomial_factorization" class="mw-redirect" title="Polynomial factorization">Polynomial factorization</a></div>

Except for factorization, all previous properties of K[X] are <a href="Effective_proof" class="mw-redirect" title="Effective proof">effective</a>, since their proofs, as sketched above, are associated with <a href="Algorithm" title="Algorithm">algorithms</a> for testing the property and computing the polynomials whose existence are asserted. Moreover these algorithms are efficient, as their <a href="Computational_complexity" title="Computational complexity">computational complexity</a> is a <a href="Quadratic_time" class="mw-redirect" title="Quadratic time">quadratic</a> function of the input size.

The situation is completely different for factorization: the proof of the unique factorization does not give any hint for a method for factorizing. Already for the integers, there is no known algorithm running on a classical (non-quantum) computer for factorizing them in <a href="Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a>. This is the basis of the <a href="RSA_cryptosystem" title="RSA cryptosystem">RSA cryptosystem</a>, widely used for secure Internet communications.

In the case of K[X], the factors, and the methods for computing them, depend strongly on K. Over the complex numbers, the irreducible factors (those that cannot be factorized further) are all of degree one, while, over the real numbers, there are irreducible polynomials of degree 2, and, over the <a href="Rational_number" title="Rational number">rational numbers</a>, there are irreducible polynomials of any degree. For example, the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{4}-2}">

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<annotation encoding="application/x-tex">{\displaystyle (X-{\sqrt[{4}]{2}})(X+{\sqrt[{4}]{2}})(X-i{\sqrt[{4}]{2}})(X+i{\sqrt[{4}]{2}})}</annotation>

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</math><img src="./90eb19786144d561600d00a4ffa682c257467e94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.517ex; height:3.176ex;" alt="{\displaystyle (X-{\sqrt[{4}]{2}})(X+{\sqrt[{4}]{2}})(X-i{\sqrt[{4}]{2}})(X+i{\sqrt[{4}]{2}})}" loading="lazy"> over the complex numbers.

The existence of a factorization algorithm depends also on the ground field. In the case of the real or complex numbers, <a href="Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a> shows that the roots of some polynomials, and thus the irreducible factors, cannot be computed exactly. Therefore, a factorization algorithm can compute only approximations of the factors. Various algorithms have been designed for computing such approximations, see <a href="Root_finding_of_polynomials" class="mw-redirect" title="Root finding of polynomials">Root finding of polynomials</a>.

There is an example of a field K such that there exist exact algorithms for the arithmetic operations of K, but there cannot exist any algorithm for deciding whether a polynomial of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{p}-a}">

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<annotation encoding="application/x-tex">{\displaystyle X^{p}-a}</annotation>

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</math><img src="./f8c89727242046a8131f11cd9925bb3b32903a39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.126ex; height:2.509ex;" alt="{\displaystyle X^{p}-a}" loading="lazy"> is <a href="Irreducible_polynomial" title="Irreducible polynomial">irreducible</a> or is a product of polynomials of lower degree.<a href="#cite_note-12">[12]</a>

On the other hand, over the rational numbers and over finite fields, the situation is better than for <a href="Integer_factorization" title="Integer factorization">integer factorization</a>, as there are <a href="Factorization_of_polynomials" title="Factorization of polynomials">factorization algorithms</a> that have a <a href="Polynomial_complexity" class="mw-redirect" title="Polynomial complexity">polynomial complexity</a>. They are implemented in most general purpose <a href="Computer_algebra_system" title="Computer algebra system">computer algebra systems</a>.

<div class="mw-heading mw-heading3"><h3 id="Minimal_polynomial">Minimal polynomial</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">Minimal polynomial (field theory)</a></div>

If θ is an element of an <a href="Associative_algebra" title="Associative algebra">associative K-algebra</a> L, the <a href="#Polynomial_evaluation">polynomial evaluation</a> at θ is the unique <a href="Algebra_homomorphism" class="mw-redirect" title="Algebra homomorphism">algebra homomorphism</a> φ from K[X] into L that maps X to θ and does not affect the elements of K itself (it is the <a href="Identity_function" title="Identity function">identity map</a> on K). It consists of substituting X with θ in every polynomial. That is,

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<annotation encoding="application/x-tex">{\displaystyle \varphi \left(a_{m}X^{m}+a_{m-1}X^{m-1}+\cdots +a_{1}X+a_{0}\right)=a_{m}\theta ^{m}+a_{m-1}\theta ^{m-1}+\cdots +a_{1}\theta +a_{0}.}</annotation>

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</math><img src="./635eb6ad95e5d52ecb84dc69b026cf78ebb147c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:81.055ex; height:3.343ex;" alt="{\displaystyle \varphi \left(a_{m}X^{m}+a_{m-1}X^{m-1}+\cdots +a_{1}X+a_{0}\right)=a_{m}\theta ^{m}+a_{m-1}\theta ^{m-1}+\cdots +a_{1}\theta +a_{0}.}" loading="lazy"></dd></dl>

The image of this evaluation homomorphism is the subalgebra generated by θ, which is necessarily commutative.

If φ is injective, the subalgebra generated by θ is isomorphic to K[X]. In this case, this subalgebra is often denoted by K[θ]. The notation ambiguity is generally harmless, because of the isomorphism.

If the evaluation homomorphism is not injective, this means that its <a href="Kernel_(algebra)" title="Kernel (algebra)">kernel</a> is a nonzero <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a>, consisting of all polynomials that become zero when X is substituted with θ. This ideal consists of all multiples of some monic polynomial, that is called the minimal polynomial of θ. The term minimal is motivated by the fact that its degree is minimal among the degrees of the elements of the ideal.

There are two main cases where minimal polynomials are considered.

In <a href="Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">field theory</a> and <a href="Number_theory" title="Number theory">number theory</a>, an element θ of an <a href="Extension_field" class="mw-redirect" title="Extension field">extension field</a> L of K is <a href="Algebraic_element" title="Algebraic element">algebraic</a> over K if it is a root of some polynomial with coefficients in K. The <a href="Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">minimal polynomial</a> over K of θ is thus the monic polynomial of minimal degree that has θ as a root. Because L is a field, this minimal polynomial is necessarily <a href="Irreducible_polynomial" title="Irreducible polynomial">irreducible</a> over K. For example, the minimal polynomial (over the reals as well as over the rationals) of the <a href="Complex_number" title="Complex number">complex number</a> i is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{2}+1}">

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</math><img src="./759c679330a1c67db74a3da9ee5cca488de3a589.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.054ex; height:2.843ex;" alt="{\displaystyle X^{2}+1}" loading="lazy">. The <a href="Cyclotomic_polynomial" title="Cyclotomic polynomial">cyclotomic polynomials</a> are the minimal polynomials of the <a href="Roots_of_unity" class="mw-redirect" title="Roots of unity">roots of unity</a>.

In <a href="Linear_algebra" title="Linear algebra">linear algebra</a>, the n×n <a href="Square_matrices" class="mw-redirect" title="Square matrices">square matrices</a> over K form an <a href="Associative_algebra" title="Associative algebra">associative K-algebra</a> of finite dimension (as a vector space). Therefore the evaluation homomorphism cannot be injective, and every matrix has a <a href="Minimal_polynomial_(linear_algebra)" title="Minimal polynomial (linear algebra)">minimal polynomial</a> (not necessarily irreducible). By <a href="Cayley%E2%80%93Hamilton_theorem" title="Cayley–Hamilton theorem">Cayley–Hamilton theorem</a>, the evaluation homomorphism maps to zero the <a href="Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> of a matrix. It follows that the minimal polynomial divides the characteristic polynomial, and therefore that the degree of the minimal polynomial is at most n.

<div class="mw-heading mw-heading3"><h3 id="Quotient_ring">Quotient ring</h3></div>

In the case of K[X], the <a href="Quotient_ring" title="Quotient ring">quotient ring</a> by an ideal can be built, as in the general case, as a set of <a href="Equivalence_class" title="Equivalence class">equivalence classes</a>. However, as each equivalence class contains exactly one polynomial of minimal degree, another construction is often more convenient.

Given a polynomial p of degree d, the quotient ring of K[X] by the <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> generated by p can be identified with the <a href="Vector_space" title="Vector space">vector space</a> of the polynomials of degrees less than d, with the "multiplication modulo p" as a multiplication, the multiplication modulo p consisting of the remainder under the division by p of the (usual) product of polynomials. This quotient ring is variously denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/pK[X],}">

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<annotation encoding="application/x-tex">{\displaystyle K[X]/pK[X],}</annotation>

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</math><img src="./4e52be91379b318a4f22919604ec7f8d9f2d1e4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.658ex; height:2.843ex;" alt="{\displaystyle K[X]/pK[X],}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/\langle p\rangle ,}">

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<annotation encoding="application/x-tex">{\displaystyle K[X]/\langle p\rangle ,}</annotation>

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</math><img src="./337bc88869134a1aeba2c8700c0b6a60aafdd362.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.128ex; height:2.843ex;" alt="{\displaystyle K[X]/\langle p\rangle ,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/(p),}">

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<annotation encoding="application/x-tex">{\displaystyle K[X]/(p),}</annotation>

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</math><img src="./94a047c03e08df13a7edb8c084cac66aca071e8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.128ex; height:2.843ex;" alt="{\displaystyle K[X]/(p),}" loading="lazy"> or simply <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/p.}">

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<annotation encoding="application/x-tex">{\displaystyle K[X]/p.}</annotation>

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</math><img src="./e7a2a61ff184564b687382271d277951d09cf149.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.318ex; height:2.843ex;" alt="{\displaystyle K[X]/p.}" loading="lazy">

The ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/(p)}">

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<annotation encoding="application/x-tex">{\displaystyle K[X]/(p)}</annotation>

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</math><img src="./bdf01b3af7443f2d6228b55d165abf44f9febf70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.481ex; height:2.843ex;" alt="{\displaystyle K[X]/(p)}" loading="lazy"> is a field if and only if p is an <a href="Irreducible_polynomial" title="Irreducible polynomial">irreducible polynomial</a>. In fact, if p is irreducible, every nonzero polynomial q of lower degree is coprime with p, and <a href="B%C3%A9zout's_identity" title="Bézout's identity">Bézout's identity</a> allows computing r and s such that sp + qr = 1; so, r is the <a href="Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> of q modulo p. Conversely, if p is reducible, then there exist polynomials a, b of degrees lower than deg(p) such that ab = p ; so a, b are nonzero <a href="Zero_divisor" title="Zero divisor">zero divisors</a> modulo p, and cannot be invertible.

For example, the standard definition of the field of the complex numbers can be summarized by saying that it is the quotient ring

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} =\mathbb {R} [X]/(X^{2}+1),}</annotation>

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</math><img src="./2f7203506bc7eaacd73467ad4b95bd0cdb9db0e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.401ex; height:3.176ex;" alt="{\displaystyle \mathbb {C} =\mathbb {R} [X]/(X^{2}+1),}" loading="lazy"></dd></dl>

and that the image of X in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

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</math><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"> is denoted by i. In fact, by the above description, this quotient consists of all polynomials of degree one in i, which have the form a + bi, with a and b in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} .}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} .}</annotation>

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</math><img src="./dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} .}" loading="lazy"> The remainder of the Euclidean division that is needed for multiplying two elements of the quotient ring is obtained by replacing i2 by −1 in their product as polynomials (this is exactly the usual definition of the product of complex numbers).

Let θ be an <a href="Algebraic_element" title="Algebraic element">algebraic element</a> in a K-algebra A. By algebraic, one means that θ has a minimal polynomial p. The <a href="First_ring_isomorphism_theorem" class="mw-redirect" title="First ring isomorphism theorem">first ring isomorphism theorem</a> asserts that the substitution homomorphism induces an <a href="Isomorphism" title="Isomorphism">isomorphism</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/(p)}">

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</math><img src="./bdf01b3af7443f2d6228b55d165abf44f9febf70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.481ex; height:2.843ex;" alt="{\displaystyle K[X]/(p)}" loading="lazy"> onto the image K[θ] of the substitution homomorphism. In particular, if A is a <a href="Simple_extension" title="Simple extension">simple extension</a> of K generated by θ, this allows identifying A and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/(p).}">

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</math><img src="./d2c8230122367e1a79b8989274fa065b453332d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.128ex; height:2.843ex;" alt="{\displaystyle K[X]/(p).}" loading="lazy"> This identification is widely used in <a href="Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>.

<div class="mw-heading mw-heading3"><h3 id="Modules">Modules</h3></div>

The <a href="Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain" title="Structure theorem for finitely generated modules over a principal ideal domain">structure theorem for finitely generated modules over a principal ideal domain</a> applies to

K[X], when K is a field. This means that every finitely generated module over K[X] may be decomposed into a <a href="Direct_sum" title="Direct sum">direct sum</a> of a <a href="Free_module" title="Free module">free module</a> and finitely many modules of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/\left\langle P^{k}\right\rangle }">

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</math><img src="./b6693e2e2afd887e4df88c3d377ef760d3af3033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.994ex; height:3.343ex;" alt="{\displaystyle K[X]/\left\langle P^{k}\right\rangle }" loading="lazy">, where P is an <a href="Irreducible_polynomial" title="Irreducible polynomial">irreducible polynomial</a> over K and k a positive integer.

<div class="mw-heading mw-heading2"><h2 id="Definition_(multivariate_case)">Definition (multivariate case)</h2></div>

Given n symbols <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\dots ,X_{n},}">

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</math><img src="./54d11af8d1db16f154c91c2bff262f719a8e43f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.946ex; height:2.509ex;" alt="{\displaystyle X_{1},\dots ,X_{n},}" loading="lazy"> called <a href="Indeterminate_(variable)" class="mw-redirect" title="Indeterminate (variable)">indeterminates</a>, a <a href="Monomial" title="Monomial">monomial</a> (also called power product)

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is a formal product of these indeterminates, possibly raised to a nonnegative power. As usual, exponents equal to one and factors with a zero exponent can be omitted. In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}^{0}\cdots X_{n}^{0}=1.}">

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The <a href="Tuple" title="Tuple">tuple</a> of exponents α = (α1, …, αn) is called the multidegree or exponent vector of the monomial. For a less cumbersome notation, the abbreviation

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<annotation encoding="application/x-tex">{\displaystyle X^{\alpha }=X_{1}^{\alpha _{1}}\cdots X_{n}^{\alpha _{n}}}</annotation>

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is often used. The degree of a monomial Xα, frequently denoted deg α or |α|, is the sum of its exponents:

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<annotation encoding="application/x-tex">{\displaystyle \deg \alpha =\sum _{i=1}^{n}\alpha _{i}.}</annotation>

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A polynomial in these indeterminates, with coefficients in a field K, or more generally a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>, is a finite <a href="Linear_combination" title="Linear combination">linear combination</a> of monomials

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where the coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\alpha }}">

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</math><img src="./c7c942b28683dde0b495e3ae3f700010f9e46c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.543ex; height:2.009ex;" alt="{\displaystyle p_{\alpha }}" loading="lazy"> are elements of K. The degree of a nonzero polynomial is the maximum of the degrees of its monomials with nonzero coefficients.

The set of polynomials in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\dots ,X_{n},}">

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</math><img src="./54d11af8d1db16f154c91c2bff262f719a8e43f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.946ex; height:2.509ex;" alt="{\displaystyle X_{1},\dots ,X_{n},}" loading="lazy"> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\dots ,X_{n}],}">

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<annotation encoding="application/x-tex">{\displaystyle K[X_{1},\dots ,X_{n}],}</annotation>

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</math><img src="./11a7e9bbac4ce6c4970f823182e7580f0299e952.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.306ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\dots ,X_{n}],}" loading="lazy"> is thus a <a href="Vector_space" title="Vector space">vector space</a> (or a <a href="Free_module" title="Free module">free module</a>, if K is a ring) that has the monomials as a basis.

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<annotation encoding="application/x-tex">{\displaystyle K[X_{1},\dots ,X_{n}]}</annotation>

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</math><img src="./47cf0764b4682f6c7a4827df7693feccd72ff86d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\dots ,X_{n}]}" loading="lazy"> is naturally equipped (see below) with a multiplication that makes a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>, and an <a href="Associative_algebra" title="Associative algebra">associative algebra</a> over K, called the polynomial ring in n indeterminates over K (the definite article the reflects that it is uniquely defined up to the name and the order of the indeterminates). If the ring K is <a href="Commutative_ring" title="Commutative ring">commutative</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\dots ,X_{n}]}">

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<annotation encoding="application/x-tex">{\displaystyle K[X_{1},\dots ,X_{n}]}</annotation>

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</math><img src="./47cf0764b4682f6c7a4827df7693feccd72ff86d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\dots ,X_{n}]}" loading="lazy"> is also a commutative ring.

<div class="mw-heading mw-heading3"><h3 id="Operations_in_K[X1,_...,_Xn]">Operations in K[X1, ..., Xn]</h3></div>

Addition and scalar multiplication of polynomials are those of a <a href="Vector_space" title="Vector space">vector space</a> or <a href="Free_module" title="Free module">free module</a> equipped by a specific basis (here the basis of the monomials). Explicitly, let

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<annotation encoding="application/x-tex">{\displaystyle p=\sum _{\alpha \in I}p_{\alpha }X^{\alpha },\quad q=\sum _{\beta \in J}q_{\beta }X^{\beta },}</annotation>

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</math><img src="./3b42c5122dacb316e5c124daa9ec144b1ac3ea8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.089ex; width:31.13ex; height:5.843ex;" alt="{\displaystyle p=\sum _{\alpha \in I}p_{\alpha }X^{\alpha },\quad q=\sum _{\beta \in J}q_{\beta }X^{\beta },}" loading="lazy">

where I and J are finite sets of exponent vectors.

The scalar multiplication of p and a scalar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in K}">

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<annotation encoding="application/x-tex">{\displaystyle c\in K}</annotation>

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</math><img src="./a5e894da36b4db8c32e35962d3950750ad556f46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle c\in K}" loading="lazy"> is

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<annotation encoding="application/x-tex">{\displaystyle cp=\sum _{\alpha \in I}cp_{\alpha }X^{\alpha }.}</annotation>

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</math><img src="./35593d29ee114427d3e0edd57a138158b3e3f5d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.405ex; height:5.676ex;" alt="{\displaystyle cp=\sum _{\alpha \in I}cp_{\alpha }X^{\alpha }.}" loading="lazy"></dd></dl>

The addition of p and q is

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<annotation encoding="application/x-tex">{\displaystyle p+q=\sum _{\alpha \in I\cup J}(p_{\alpha }+q_{\alpha })X^{\alpha },}</annotation>

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</math><img src="./5d65b3e661459dc2f908a0a273fd499275dbfd51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:26.733ex; height:5.676ex;" alt="{\displaystyle p+q=\sum _{\alpha \in I\cup J}(p_{\alpha }+q_{\alpha })X^{\alpha },}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\alpha }=0}">

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<annotation encoding="application/x-tex">{\displaystyle p_{\alpha }=0}</annotation>

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</math><img src="./a2139b4e777b292ab16a4e62c6de185455ee5ea2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.804ex; height:2.509ex;" alt="{\displaystyle p_{\alpha }=0}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \not \in I,}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha \not \in I,}</annotation>

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</math><img src="./97a89c2ce3f55baffe30e9cb337573faca198822.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.147ex; height:2.676ex;" alt="{\displaystyle \alpha \not \in I,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{\beta }=0}">

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<annotation encoding="application/x-tex">{\displaystyle q_{\beta }=0}</annotation>

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</math><img src="./a4cc03332c54e0b96f2d8ea6cf41877d49c3b204.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.472ex; height:2.843ex;" alt="{\displaystyle q_{\beta }=0}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \not \in J.}">

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<annotation encoding="application/x-tex">{\displaystyle \beta \not \in J.}</annotation>

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</math><img src="./e36508f9f7c96ffc2cc6363277ff6c42f4a0c5e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.291ex; height:2.676ex;" alt="{\displaystyle \beta \not \in J.}" loading="lazy"> Moreover, if one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\alpha }+q_{\alpha }=0}">

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<annotation encoding="application/x-tex">{\displaystyle p_{\alpha }+q_{\alpha }=0}</annotation>

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</math><img src="./83430a9c4c701a278bae46ef2ae8f678d3423e3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.966ex; height:2.509ex;" alt="{\displaystyle p_{\alpha }+q_{\alpha }=0}" loading="lazy"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in I\cap J,}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha \in I\cap J,}</annotation>

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</math><img src="./61388279ee3615b096e1308a624539ce723f1647.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.201ex; height:2.509ex;" alt="{\displaystyle \alpha \in I\cap J,}" loading="lazy"> the corresponding zero term is removed from the result.

The multiplication is

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<annotation encoding="application/x-tex">{\displaystyle pq=\sum _{\gamma \in I+J}\left(\sum _{\alpha ,\beta \mid \alpha +\beta =\gamma }p_{\alpha }q_{\beta }\right)X^{\gamma },}</annotation>

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</math><img src="./ef7dc88dfeec750181bcf03833079695e5774efd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; margin-left: -0.089ex; width:32.577ex; height:8.176ex;" alt="{\displaystyle pq=\sum _{\gamma \in I+J}\left(\sum _{\alpha ,\beta \mid \alpha +\beta =\gamma }p_{\alpha }q_{\beta }\right)X^{\gamma },}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I+J}">

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<annotation encoding="application/x-tex">{\displaystyle I+J}</annotation>

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</math><img src="./ff0e36f3880c3eae9b440c7e412a0f02664b8624.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.484ex; height:2.343ex;" alt="{\displaystyle I+J}" loading="lazy"> is the set of the sums of one exponent vector in I and one other in J (usual sum of vectors). In particular, the product of two monomials is a monomial whose exponent vector is the sum of the exponent vectors of the factors.

The verification of the axioms of an <a href="Associative_algebra" title="Associative algebra">associative algebra</a> is straightforward.

<div class="mw-heading mw-heading3"><h3 id="Polynomial_expression">Polynomial expression</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Algebraic_expression" title="Algebraic expression">Algebraic expression</a></div>

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A polynomial expression is an <a href="Expression_(mathematics)" title="Expression (mathematics)">expression</a> built with scalars (elements of K), indeterminates, and the operators of addition, multiplication, and exponentiation to nonnegative integer powers.

As all these operations are defined in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\dots ,X_{n}]}">

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<annotation encoding="application/x-tex">{\displaystyle K[X_{1},\dots ,X_{n}]}</annotation>

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</math><img src="./47cf0764b4682f6c7a4827df7693feccd72ff86d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\dots ,X_{n}]}" loading="lazy"> a polynomial expression represents a polynomial, that is an element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\dots ,X_{n}].}">

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<annotation encoding="application/x-tex">{\displaystyle K[X_{1},\dots ,X_{n}].}</annotation>

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</math><img src="./aa5971b61b083b8320deec96bd98af348bb33c48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.306ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\dots ,X_{n}].}" loading="lazy"> The definition of a polynomial as a linear combination of monomials is a particular polynomial expression, which is often called the canonical form, normal form, or expanded form of the polynomial.

Given a polynomial expression, one can compute the expanded form of the represented polynomial by expanding with the <a href="Distributive_law" class="mw-redirect" title="Distributive law">distributive law</a> all the products that have a sum among their factors, and then using <a href="Commutativity" class="mw-redirect" title="Commutativity">commutativity</a> (except for the product of two scalars), and <a href="Associativity" class="mw-redirect" title="Associativity">associativity</a> for transforming the terms of the resulting sum into products of a scalar and a monomial; then one gets the canonical form by regrouping the <a href="Like_terms" title="Like terms">like terms</a>.

The distinction between a polynomial expression and the polynomial that it represents is relatively recent, and mainly motivated by the rise of <a href="Computer_algebra" title="Computer algebra">computer algebra</a>, where, for example, the test whether two polynomial expressions represent the same polynomial may be a nontrivial computation.

<div class="mw-heading mw-heading3"><h3 id="Categorical_characterization">Categorical characterization</h3></div>

If K is a commutative ring, the polynomial ring K[X1, …, Xn] has the following <a href="Universal_property" title="Universal property">universal property</a>: for every <a href="Commutative_algebra_(structure)" class="mw-redirect" title="Commutative algebra (structure)">commutative K-algebra</a> A, and every n-<a href="Tuple" title="Tuple">tuple</a> (x1, …, xn) of elements of A, there is a unique <a href="Algebra_homomorphism" class="mw-redirect" title="Algebra homomorphism">algebra homomorphism</a> from K[X1, …, Xn] to A that maps each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>

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</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"> to the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}.}">

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<annotation encoding="application/x-tex">{\displaystyle x_{i}.}</annotation>

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</math><img src="./ca686849ad350b47ebab4a394b66d84d5fa942bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.776ex; height:2.009ex;" alt="{\displaystyle x_{i}.}" loading="lazy"> This homomorphism is the evaluation homomorphism that consists in substituting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>

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</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

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</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> in every polynomial.

As it is the case for every universal property, this characterizes the pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K[X_{1},\dots ,X_{n}],(X_{1},\dots ,X_{n}))}">

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<annotation encoding="application/x-tex">{\displaystyle (K[X_{1},\dots ,X_{n}],(X_{1},\dots ,X_{n}))}</annotation>

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</math><img src="./7aa09822e563058b767425df5abf0097dffe1b16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.611ex; height:2.843ex;" alt="{\displaystyle (K[X_{1},\dots ,X_{n}],(X_{1},\dots ,X_{n}))}" loading="lazy"> up to a unique <a href="Isomorphism" title="Isomorphism">isomorphism</a>.

This may also be interpreted in terms of <a href="Adjoint_functor" class="mw-redirect" title="Adjoint functor">adjoint functors</a>. More precisely, let SET and ALG be respectively the <a href="Category_(mathematics)" title="Category (mathematics)">categories</a> of sets and commutative K-algebras (here, and in the following, the morphisms are trivially defined). There is a <a href="Forgetful_functor" title="Forgetful functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {F} :\mathrm {ALG} \to \mathrm {SET} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {F} :\mathrm {ALG} \to \mathrm {SET} }</annotation>

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</math><img src="./e37f477291831ab140c086d3d7ba63e73eee6629.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.643ex; height:2.176ex;" alt="{\displaystyle \mathrm {F} :\mathrm {ALG} \to \mathrm {SET} }" loading="lazy"> that maps algebras to their underlying sets. On the other hand, the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\mapsto K[X]}">

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<annotation encoding="application/x-tex">{\displaystyle X\mapsto K[X]}</annotation>

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</math><img src="./56dfb938869f70cb50f04d3ec6e66fceebe3b9d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.934ex; height:2.843ex;" alt="{\displaystyle X\mapsto K[X]}" loading="lazy"> defines a functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {POL} :\mathrm {SET} \to \mathrm {ALG} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {POL} :\mathrm {SET} \to \mathrm {ALG} }</annotation>

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</math><img src="./1f76a3cd36649e049d78f5f730aa4bc09ada0189.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.969ex; height:2.176ex;" alt="{\displaystyle \mathrm {POL} :\mathrm {SET} \to \mathrm {ALG} }" loading="lazy"> in the other direction. (If X is infinite, K[X] is the set of all polynomials in a finite number of elements of X.)

The universal property of the polynomial ring means that F and POL are <a href="Adjoint_functors" title="Adjoint functors">adjoint functors</a>. That is, there is a bijection

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{\mathrm {SET} }(X,\operatorname {F} (A))\cong \operatorname {Hom} _{\mathrm {ALG} }(K[X],A).}</annotation>

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</math><img src="./3587f518c9c731771ac7fbea5b3aefcfa747ad1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.482ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} _{\mathrm {SET} }(X,\operatorname {F} (A))\cong \operatorname {Hom} _{\mathrm {ALG} }(K[X],A).}" loading="lazy"></dd></dl>

This may be expressed also by saying that polynomial rings are free commutative algebras, since they are <a href="Free_object" title="Free object">free objects</a> in the category of commutative algebras. Similarly, a polynomial ring with integer coefficients is the free commutative ring over its set of variables, since commutative rings and commutative algebras over the integers are the same thing.

<div class="mw-heading mw-heading2"><h2 id="Graded_structure">Graded structure</h2></div>

Every polynomial ring is a <a href="Graded_ring" title="Graded ring">graded ring</a>: one can write the polynomial ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=K[X_{1},\ldots ,X_{n}]}">

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<annotation encoding="application/x-tex">{\displaystyle R=K[X_{1},\ldots ,X_{n}]}</annotation>

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</math><img src="./543d3f2ccc11434e73b61ae8ddb83d406476e02d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.522ex; height:2.843ex;" alt="{\displaystyle R=K[X_{1},\ldots ,X_{n}]}" loading="lazy"> as a <a href="Direct_sum" title="Direct sum">direct sum</a>

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<annotation encoding="application/x-tex">{\displaystyle R=\bigoplus _{i=0}^{\infty }R_{i}}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{i}}</annotation>

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</math><img src="./db421291be9d0103404ced7495b363437b67b6b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.564ex; height:2.509ex;" alt="{\displaystyle R_{i}}" loading="lazy"> is the subspace consisting of all <a href="Homogeneous_polynomial" title="Homogeneous polynomial">homogeneous polynomials</a> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> (along with the zero polynomial); then for any elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in R_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle f\in R_{i}}</annotation>

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</math><img src="./908d269580f95277655f88abf37452b3e9fe40b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.683ex; height:2.509ex;" alt="{\displaystyle f\in R_{i}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in R_{j}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle g\in R_{j}}</annotation>

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</math><img src="./6c5db84994a298fb995314a429f1fdb44dbb429e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.63ex; height:2.843ex;" alt="{\displaystyle g\in R_{j}}" loading="lazy">, their product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fg}">

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<annotation encoding="application/x-tex">{\displaystyle fg}</annotation>

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</math><img src="./06bac4638bb56f14688118ce88c188c7a021eb29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.395ex; height:2.509ex;" alt="{\displaystyle fg}" loading="lazy"> belongs to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{i+j}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{i+j}}</annotation>

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</math><img src="./d2e0ad2669f54bb8cc42e97cd3fd546e0cf5ad21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.52ex; height:2.843ex;" alt="{\displaystyle R_{i+j}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Univariate_over_a_ring_vs._multivariate">Univariate over a ring vs. multivariate</h2></div>

A polynomial in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]}">

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<annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]}</annotation>

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</math><img src="./3d1ac7a7496de3efbb2cd22c42398736c86a2b48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]}" loading="lazy"> can be considered as a univariate polynomial in the indeterminate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation>

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</math><img src="./72a8564cedc659cf2f95ae68bc5de2f5207a3285.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}" loading="lazy"> over the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n-1}],}">

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<annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n-1}],}</annotation>

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</math><img src="./726494480638e29f49e3999e1f5c2d4d99d51933.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.406ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n-1}],}" loading="lazy"> by regrouping the terms that contain the same power of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n},}">

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<annotation encoding="application/x-tex">{\displaystyle X_{n},}</annotation>

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</math><img src="./6759dac8dbc82523fd14db264c829247879174b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.79ex; height:2.509ex;" alt="{\displaystyle X_{n},}" loading="lazy"> that is, by using the identity

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<annotation encoding="application/x-tex">{\displaystyle \sum _{(\alpha _{1},\ldots ,\alpha _{n})\in I}c_{\alpha _{1},\ldots ,\alpha _{n}}X_{1}^{\alpha _{1}}\cdots X_{n}^{\alpha _{n}}=\sum _{i}\left(\sum _{(\alpha _{1},\ldots ,\alpha _{n-1})\mid (\alpha _{1},\ldots ,\alpha _{n-1},i)\in I}c_{\alpha _{1},\ldots ,\alpha _{n-1}}X_{1}^{\alpha _{1}}\cdots X_{n-1}^{\alpha _{n-1}}\right)X_{n}^{i},}</annotation>

</semantics>

</math><img src="./cf6c95b27250e5a5cf12475486e1aae8a5b84b96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:91.743ex; height:8.343ex;" alt="{\displaystyle \sum _{(\alpha _{1},\ldots ,\alpha _{n})\in I}c_{\alpha _{1},\ldots ,\alpha _{n}}X_{1}^{\alpha _{1}}\cdots X_{n}^{\alpha _{n}}=\sum _{i}\left(\sum _{(\alpha _{1},\ldots ,\alpha _{n-1})\mid (\alpha _{1},\ldots ,\alpha _{n-1},i)\in I}c_{\alpha _{1},\ldots ,\alpha _{n-1}}X_{1}^{\alpha _{1}}\cdots X_{n-1}^{\alpha _{n-1}}\right)X_{n}^{i},}" loading="lazy"></dd></dl>

which results from the distributivity and associativity of ring operations.

This means that one has an <a href="Algebra_isomorphism" class="mw-redirect" title="Algebra isomorphism">algebra isomorphism</a>

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<annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]\cong (K[X_{1},\ldots ,X_{n-1}])[X_{n}]}</annotation>

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</math><img src="./a3c3ee9af59d2086949741d7059b6a2f3ddcb4ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.763ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]\cong (K[X_{1},\ldots ,X_{n-1}])[X_{n}]}" loading="lazy"></dd></dl>

that maps each indeterminate to itself. (This isomorphism is often written as an equality, which is justified by the fact that polynomial rings are defined up to a unique isomorphism.)

In other words, a multivariate polynomial ring can be considered as a univariate polynomial over a smaller polynomial ring. This is commonly used for proving properties of multivariate polynomial rings, by <a href="Mathematical_induction" title="Mathematical induction">induction</a> on the number of indeterminates.

The main such properties are listed below.

<div class="mw-heading mw-heading3"><h3 id="Properties_that_pass_from_R_to_R[X]">Properties that pass from R to R[X]</h3></div>

In this section, R is a commutative ring, K is a field, X denotes a single indeterminate, and, as usual, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

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</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"> is the ring of integers. Here is the list of the main ring properties that remain true when passing from R to R[X].

<ul><li>If R is an <a href="Integral_domain" title="Integral domain">integral domain</a> then the same holds for R[X] (since the leading coefficient of a product of polynomials is, if not zero, the product of the leading coefficients of the factors).

<ul><li>In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]}">

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]}</annotation>

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</math><img src="./3d1ac7a7496de3efbb2cd22c42398736c86a2b48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}</annotation>

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</math><img src="./90d22cfd2dfdbfee457e462397218f0845d1d5c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.143ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}" loading="lazy"> are integral domains.</li></ul></li>

<li>If R is a <a href="Unique_factorization_domain" title="Unique factorization domain">unique factorization domain</a> then the same holds for R[X]. This results from <a href="Gauss's_lemma_(polynomial)" class="mw-redirect" title="Gauss's lemma (polynomial)">Gauss's lemma</a> and the unique factorization property of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L[X],}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle L[X],}</annotation>

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</math><img src="./6b467e0e9a96b2086bcee9943ab935fd960491a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.503ex; height:2.843ex;" alt="{\displaystyle L[X],}" loading="lazy"> where L is the field of fractions of R.

<ul><li>In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]}</annotation>

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</math><img src="./3d1ac7a7496de3efbb2cd22c42398736c86a2b48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}">

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}</annotation>

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</math><img src="./90d22cfd2dfdbfee457e462397218f0845d1d5c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.143ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}" loading="lazy"> are unique factorization domains.</li></ul></li>

<li>If R is a <a href="Noetherian_ring" title="Noetherian ring">Noetherian ring</a>, then the same holds for R[X].

<ul><li>In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]}</annotation>

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</math><img src="./3d1ac7a7496de3efbb2cd22c42398736c86a2b48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}">

</mrow>

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}</annotation>

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</math><img src="./90d22cfd2dfdbfee457e462397218f0845d1d5c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.143ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}" loading="lazy"> are Noetherian rings; this is <a href="Hilbert's_basis_theorem" title="Hilbert's basis theorem">Hilbert's basis theorem</a>.</li></ul></li>

<li>If R is a Noetherian ring, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim R[X]=1+\dim R,}">

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<annotation encoding="application/x-tex">{\displaystyle \dim R[X]=1+\dim R,}</annotation>

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</math><img src="./4262e007a0121c85eb2ae8f6e3a2165b55691b02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.075ex; height:2.843ex;" alt="{\displaystyle \dim R[X]=1+\dim R,}" loading="lazy"> where "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim }">

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<annotation encoding="application/x-tex">{\displaystyle \dim }</annotation>

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</math><img src="./66115c83c4bb19068adb45849f0f596647c18f2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.875ex; height:2.176ex;" alt="{\displaystyle \dim }" loading="lazy">" denotes the <a href="Krull_dimension" title="Krull dimension">Krull dimension</a>.

<ul><li>In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim K[X_{1},\ldots ,X_{n}]=n}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \dim K[X_{1},\ldots ,X_{n}]=n}</annotation>

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</math><img src="./2a255aafe61b6fcedfdc73a8e0e7368c5ad0129c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.415ex; height:2.843ex;" alt="{\displaystyle \dim K[X_{1},\ldots ,X_{n}]=n}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim \mathbb {Z} [X_{1},\ldots ,X_{n}]=n+1.}">

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<annotation encoding="application/x-tex">{\displaystyle \dim \mathbb {Z} [X_{1},\ldots ,X_{n}]=n+1.}</annotation>

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</math><img src="./646107e48d7f66b4426d2792c33a718cefcbae19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.549ex; height:2.843ex;" alt="{\displaystyle \dim \mathbb {Z} [X_{1},\ldots ,X_{n}]=n+1.}" loading="lazy"></li></ul></li>

<li>If R is a <a href="Regular_ring" class="mw-redirect" title="Regular ring">regular ring</a>, then the same holds for R[X]; in this case, one has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gl} \,\dim R[X]=\dim R[X]=1+\operatorname {gl} \,\dim R=1+\dim R,}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {gl} \,\dim R[X]=\dim R[X]=1+\operatorname {gl} \,\dim R=1+\dim R,}</annotation>

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</math> where "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gl} \,\dim }">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {gl} \,\dim }</annotation>

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</math><img src="./b04b60de26f746c6abd1715d4c6eec823b523424.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.459ex; height:2.509ex;" alt="{\displaystyle \operatorname {gl} \,\dim }" loading="lazy">" denotes the <a href="Global_dimension" title="Global dimension">global dimension</a>.

<ul><li>In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]}</annotation>

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</math><img src="./3d1ac7a7496de3efbb2cd22c42398736c86a2b48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}</annotation>

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</math><img src="./90d22cfd2dfdbfee457e462397218f0845d1d5c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.143ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}" loading="lazy"> are regular rings, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gl} \,\dim \mathbb {Z} [X_{1},\ldots ,X_{n}]=n+1,}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {gl} \,\dim \mathbb {Z} [X_{1},\ldots ,X_{n}]=n+1,}</annotation>

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</math><img src="./832c2bba7901e808089559e18a8646a239219853.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.132ex; height:2.843ex;" alt="{\displaystyle \operatorname {gl} \,\dim \mathbb {Z} [X_{1},\ldots ,X_{n}]=n+1,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gl} \,\dim K[X_{1},\ldots ,X_{n}]=n.}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {gl} \,\dim K[X_{1},\ldots ,X_{n}]=n.}</annotation>

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</math><img src="./de3c6e7e07280534e37854b6d8902a999f5f75fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.645ex; height:2.843ex;" alt="{\displaystyle \operatorname {gl} \,\dim K[X_{1},\ldots ,X_{n}]=n.}" loading="lazy"> The latter equality is <a href="Hilbert's_syzygy_theorem" title="Hilbert's syzygy theorem">Hilbert's syzygy theorem</a>.</li></ul></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Several_indeterminates_over_a_field">Several indeterminates over a field</h2></div>

Polynomial rings in several variables over a field are fundamental in <a href="Invariant_theory" title="Invariant theory">invariant theory</a> and <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. Some of their properties, such as those described above can be reduced to the case of a single indeterminate, but this is not always the case. In particular, because of the geometric applications, many interesting properties must be invariant under <a href="Affine_transformation" title="Affine transformation">affine</a> or <a href="Projective_transformation" class="mw-redirect" title="Projective transformation">projective</a> transformations of the indeterminates. This often implies that one cannot select one of the indeterminates for a recurrence on the indeterminates.

<a href="B%C3%A9zout's_theorem" title="Bézout's theorem">Bézout's theorem</a>, <a href="Hilbert's_Nullstellensatz" title="Hilbert's Nullstellensatz">Hilbert's Nullstellensatz</a> and <a href="Jacobian_conjecture" title="Jacobian conjecture">Jacobian conjecture</a> are among the most famous properties that are specific to multivariate polynomials over a field.

<div class="mw-heading mw-heading3"><h3 id="Hilbert's_Nullstellensatz">Hilbert's Nullstellensatz</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Hilbert's_Nullstellensatz" title="Hilbert's Nullstellensatz">Hilbert's Nullstellensatz</a></div>

The Nullstellensatz (German for "zero-locus theorem") is a theorem, first proved by <a href="David_Hilbert" title="David Hilbert">David Hilbert</a>, which extends to the multivariate case some aspects of the <a href="Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a>. It is foundational for <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, as establishing a strong link between the algebraic properties of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]}">

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<annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]}</annotation>

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</math><img src="./3d1ac7a7496de3efbb2cd22c42398736c86a2b48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]}" loading="lazy"> and the geometric properties of <a href="Algebraic_varieties" class="mw-redirect" title="Algebraic varieties">algebraic varieties</a>, that are (roughly speaking) set of points defined by <a href="Implicit_equation" class="mw-redirect" title="Implicit equation">implicit polynomial equations</a>.

The Nullstellensatz, has three main versions, each being a corollary of any other. Two of these versions are given below. For the third version, the reader is referred to the main article on the Nullstellensatz.

The first version generalizes the fact that a nonzero univariate polynomial has a <a href="Complex_number" title="Complex number">complex</a> zero if and only if it is not a constant. The statement is: a set of polynomials S in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]}">

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<annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]}</annotation>

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</math><img src="./3d1ac7a7496de3efbb2cd22c42398736c86a2b48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]}" loading="lazy"> has a common zero in an <a href="Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a> containing K, if and only if 1 does not belong to the <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> generated by S, that is, if 1 is not a <a href="Linear_combination" title="Linear combination">linear combination</a> of elements of S with polynomial coefficients.

The second version generalizes the fact that the <a href="Irreducible_polynomial" title="Irreducible polynomial">irreducible univariate polynomials</a> over the complex numbers are <a href="Associate_elements" class="mw-redirect" title="Associate elements">associate</a> to a polynomial of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X-\alpha .}">

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<annotation encoding="application/x-tex">{\displaystyle X-\alpha .}</annotation>

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</math><img src="./b5b2f3c4393953f5499cd97fe5bf1a0fcc32e77f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.955ex; height:2.343ex;" alt="{\displaystyle X-\alpha .}" loading="lazy"> The statement is: If K is algebraically closed, then the <a href="Maximal_ideal" title="Maximal ideal">maximal ideals</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]}">

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<annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]}</annotation>

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</math><img src="./3d1ac7a7496de3efbb2cd22c42398736c86a2b48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]}" loading="lazy"> have the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle X_{1}-\alpha _{1},\ldots ,X_{n}-\alpha _{n}\rangle .}">

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<annotation encoding="application/x-tex">{\displaystyle \langle X_{1}-\alpha _{1},\ldots ,X_{n}-\alpha _{n}\rangle .}</annotation>

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</math><img src="./ee4cb14f681920246a23d9ab30e94be08bc512ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.684ex; height:2.843ex;" alt="{\displaystyle \langle X_{1}-\alpha _{1},\ldots ,X_{n}-\alpha _{n}\rangle .}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Bézout's_theorem">Bézout's theorem</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="B%C3%A9zout's_theorem" title="Bézout's theorem">Bézout's theorem</a></div>

Bézout's theorem may be viewed as a multivariate generalization of the version of the <a href="Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a> that asserts that a univariate polynomial of degree n has n complex roots, if they are counted with their multiplicities.

In the case of <a href="Bivariate_polynomial" class="mw-redirect" title="Bivariate polynomial">bivariate polynomials</a>, it states that two polynomials of degrees d and e in two variables, which have no common factors of positive degree, have exactly de common zeros in an <a href="Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a> containing the coefficients, if the zeros are counted with their multiplicity and include the <a href="Point_at_infinity" title="Point at infinity">zeros at infinity</a>.

For stating the general case, and not considering "zero at infinity" as special zeros, it is convenient to work with <a href="Homogeneous_polynomial" title="Homogeneous polynomial">homogeneous polynomials</a>, and consider zeros in a <a href="Projective_space" title="Projective space">projective space</a>. In this context, a projective zero of a homogeneous polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X_{0},\ldots ,X_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle P(X_{0},\ldots ,X_{n})}</annotation>

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</math><img src="./aa61a5b87352b04e7fa758f42ac9c7bb30de899a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.854ex; height:2.843ex;" alt="{\displaystyle P(X_{0},\ldots ,X_{n})}" loading="lazy"> is, up to a scaling, a (n + 1)-<a href="Tuple" title="Tuple">tuple</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},\ldots ,x_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{0},\ldots ,x_{n})}</annotation>

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</math><img src="./d8811e57cd4d3e4fe4e7e2af3a9822f77746e8ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle (x_{0},\ldots ,x_{n})}" loading="lazy"> of elements of K that is different from (0, …, 0), and such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x_{0},\ldots ,x_{n})=0}">

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<annotation encoding="application/x-tex">{\displaystyle P(x_{0},\ldots ,x_{n})=0}</annotation>

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</math><img src="./78210f993473f3b83a6efb84e5488cd035e3ca1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.926ex; height:2.843ex;" alt="{\displaystyle P(x_{0},\ldots ,x_{n})=0}" loading="lazy">. Here, "up to a scaling" means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},\ldots ,x_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{0},\ldots ,x_{n})}</annotation>

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</math><img src="./d8811e57cd4d3e4fe4e7e2af3a9822f77746e8ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle (x_{0},\ldots ,x_{n})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda x_{0},\ldots ,\lambda x_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle (\lambda x_{0},\ldots ,\lambda x_{n})}</annotation>

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</math><img src="./4de55ec422915084a176d2b9ea56002b81e63e15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.63ex; height:2.843ex;" alt="{\displaystyle (\lambda x_{0},\ldots ,\lambda x_{n})}" loading="lazy"> are considered as the same zero for any nonzero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in K.}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda \in K.}</annotation>

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</math><img src="./2edb7ffb8ac74ac2b7b5f4c9bf4108099134aebe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.909ex; height:2.176ex;" alt="{\displaystyle \lambda \in K.}" loading="lazy"> In other words, a zero is a set of <a href="Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a> of a point in a projective space of dimension n.

Then, Bézout's theorem states: Given n homogeneous polynomials of degrees <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{1},\ldots ,d_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle d_{1},\ldots ,d_{n}}</annotation>

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</math><img src="./af2b00c84940b9421f7f66f4b7df4c59f3871068.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.869ex; height:2.509ex;" alt="{\displaystyle d_{1},\ldots ,d_{n}}" loading="lazy"> in n + 1 indeterminates, which have only a finite number of common projective zeros in an <a href="Algebraically_closed_extension" class="mw-redirect" title="Algebraically closed extension">algebraically closed extension</a> of K, the sum of the <a href="Multiplicity_(mathematics)#Intersection_multipliicty" title="Multiplicity (mathematics)">multiplicities</a> of these zeros is the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{1}\cdots d_{n}.}">

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<annotation encoding="application/x-tex">{\displaystyle d_{1}\cdots d_{n}.}</annotation>

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</math><img src="./e4b9071a2af273bee9c725c895ae394ed2b45e50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.835ex; height:2.509ex;" alt="{\displaystyle d_{1}\cdots d_{n}.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Jacobian_conjecture">Jacobian conjecture</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Jacobian_conjecture" title="Jacobian conjecture">Jacobian conjecture</a></div>

<div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2></div>

Polynomial rings can be generalized in a great many ways, including polynomial rings with generalized exponents, power series rings, <a href="Noncommutative_polynomial_ring" class="mw-redirect" title="Noncommutative polynomial ring">noncommutative polynomial rings</a>, <a href="Skew_polynomial_ring" class="mw-redirect" title="Skew polynomial ring">skew polynomial rings</a>, and polynomial <a href="Rig_(mathematics)" class="mw-redirect" title="Rig (mathematics)">rigs</a>.

<div class="mw-heading mw-heading3"><h3 id="Infinitely_many_variables">Infinitely many variables</h3></div>

One slight generalization of polynomial rings is to allow for infinitely many indeterminates. Each monomial still involves only a finite number of indeterminates (so that its degree remains finite), and each polynomial is a still a (finite) linear combination of monomials. Thus, any individual polynomial involves only finitely many indeterminates, and any finite computation involving polynomials remains inside some subring of polynomials in finitely many indeterminates. This generalization has the same property of usual polynomial rings, of being the <a href="Free_commutative_algebra" class="mw-redirect" title="Free commutative algebra">free commutative algebra</a>, the only difference is that it is a <a href="Free_object" title="Free object">free object</a> over an infinite set.

One can also consider a strictly larger ring, by defining as a generalized polynomial an infinite (or finite) formal sum of monomials with a bounded degree. This ring is larger than the usual polynomial ring, as it includes infinite sums of variables. However, it is smaller than the <a href="Power_series_ring" class="mw-redirect" title="Power series ring">ring of power series in infinitely many variables</a>. Such a ring is used for constructing the <a href="Ring_of_symmetric_functions" title="Ring of symmetric functions">ring of symmetric functions</a> over an infinite set.

<div class="mw-heading mw-heading3"><h3 id="Generalized_exponents">Generalized exponents</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Monoid_ring" title="Monoid ring">Monoid ring</a></div>

A simple generalization only changes the set from which the exponents on the variable are drawn. The formulas for addition and multiplication make sense as long as one can add exponents: Xi ⋅ Xj = Xi+j. A set for which addition makes sense (is closed and associative) is called a <a href="Monoid" title="Monoid">monoid</a>. The set of functions from a monoid N to a ring R which are nonzero at only finitely many places can be given the structure of a ring known as R[N], the monoid ring of N with coefficients in R. The addition is defined component-wise, so that if c = a + b, then cn = an + bn for every n in N. The multiplication is defined as the Cauchy product, so that if c = a ⋅ b, then for each n in N, cn is the sum of all aibj where i, j range over all pairs of elements of N which sum to n.

When N is commutative, it is convenient to denote the function a in R[N] as the formal sum:

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n\in N}a_{n}X^{n}}</annotation>

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</math><img src="./2a018e2f129e085ac7767539c233f8b4dd7076c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:9.592ex; height:5.676ex;" alt="{\displaystyle \sum _{n\in N}a_{n}X^{n}}" loading="lazy"></dd></dl>

and then the formulas for addition and multiplication are the familiar:

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<annotation encoding="application/x-tex">{\displaystyle \left(\sum _{n\in N}a_{n}X^{n}\right)+\left(\sum _{n\in N}b_{n}X^{n}\right)=\sum _{n\in N}\left(a_{n}+b_{n}\right)X^{n}}</annotation>

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</math><img src="./eec013772477be911637a9633a2f2b75df882693.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:49.099ex; height:7.509ex;" alt="{\displaystyle \left(\sum _{n\in N}a_{n}X^{n}\right)+\left(\sum _{n\in N}b_{n}X^{n}\right)=\sum _{n\in N}\left(a_{n}+b_{n}\right)X^{n}}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle \left(\sum _{n\in N}a_{n}X^{n}\right)\cdot \left(\sum _{n\in N}b_{n}X^{n}\right)=\sum _{n\in N}\left(\sum _{i+j=n}a_{i}b_{j}\right)X^{n}}</annotation>

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</math><img src="./567953e4cbcc5b43c303bc32799dc25d5a9e22e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:51.417ex; height:7.676ex;" alt="{\displaystyle \left(\sum _{n\in N}a_{n}X^{n}\right)\cdot \left(\sum _{n\in N}b_{n}X^{n}\right)=\sum _{n\in N}\left(\sum _{i+j=n}a_{i}b_{j}\right)X^{n}}" loading="lazy"></dd></dl>

where the latter sum is taken over all i, j in N that sum to n.

Some authors such as (<a href="#CITEREFLang2002">Lang 2002</a>, II,§3) go so far as to take this monoid definition as the starting point, and regular single variable polynomials are the special case where N is the monoid of non-negative integers. Polynomials in several variables simply take N to be the direct product of several copies of the monoid of non-negative integers.

Several interesting examples of rings and groups are formed by taking N to be the additive monoid of non-negative rational numbers, (<a href="#CITEREFOsborne2000">Osborne 2000</a>, §4.4). See also <a href="Puiseux_series" title="Puiseux series">Puiseux series</a>.

<div class="mw-heading mw-heading3"><h3 id="Power_series">Power series</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Formal_power_series" title="Formal power series">Formal power series</a></div>

Power series generalize the choice of exponent in a different direction by allowing infinitely many nonzero terms. This requires various hypotheses on the monoid N used for the exponents, to ensure that the sums in the Cauchy product are finite sums. Alternatively, a topology can be placed on the ring, and then one restricts to convergent infinite sums. For the standard choice of N, the non-negative integers, there is no trouble, and the ring of formal power series is defined as the set of functions from N to a ring R with addition component-wise, and multiplication given by the Cauchy product. The ring of power series can also be seen as the <a href="Completion_of_a_ring" title="Completion of a ring">ring completion</a> of the polynomial ring with respect to the ideal generated by x.

<div class="mw-heading mw-heading3"><h3 id="Noncommutative_polynomial_rings">Noncommutative polynomial rings</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Free_algebra" title="Free algebra">Free algebra</a></div>

For polynomial rings of more than one variable, the products X⋅Y and Y⋅X are simply defined to be equal. A more general notion of polynomial ring is obtained when the distinction between these two formal products is maintained. Formally, the polynomial ring in n noncommuting variables with coefficients in the ring R is the <a href="Monoid_ring" title="Monoid ring">monoid ring</a> R[N], where the monoid N is the <a href="Free_monoid" title="Free monoid">free monoid</a> on n letters, also known as the set of all strings over an alphabet of n symbols, with multiplication given by concatenation. Neither the coefficients nor the variables need commute amongst themselves, but the coefficients and variables commute with each other.

Just as the polynomial ring in n variables with coefficients in the commutative ring R is the free commutative R-algebra of rank n, the noncommutative polynomial ring in n variables with coefficients in the commutative ring R is the free associative, unital R-algebra on n generators, which is noncommutative when n > 1.

<div class="mw-heading mw-heading3"><h3 id="Differential_and_skew-polynomial_rings">Differential and skew-polynomial rings</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Ore_extension" title="Ore extension">Ore extension</a></div>

Other generalizations of polynomials are differential and skew-polynomial rings.

A differential polynomial ring is a ring of <a href="Differential_operator" title="Differential operator">differential operators</a> formed from a ring R and a <a href="Derivation_(abstract_algebra)" class="mw-redirect" title="Derivation (abstract algebra)">derivation</a> δ of R into R. This derivation operates on R, and will be denoted X, when viewed as an operator. The elements of R also operate on R by multiplication. The <a href="Function_composition" title="Function composition">composition of operators</a> is denoted as the usual multiplication. It follows that the relation δ(ab) = aδ(b) + δ(a)b may be rewritten

as

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This relation may be extended to define a skew multiplication between two polynomials in X with coefficients in R, which make them a <a href="Noncommutative_ring" title="Noncommutative ring">noncommutative ring</a>.

The standard example, called a <a href="Weyl_algebra" title="Weyl algebra">Weyl algebra</a>, takes R to be a (usual) polynomial ring k[Y ], and δ to be the standard polynomial derivative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\partial }{\partial Y}}}">

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</semantics>

</math><img src="./e9fd20891fa74c1fd9c2d50c62ed85f91e3f6026.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.022ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\partial }{\partial Y}}}" loading="lazy">. Taking a = Y in the above relation, one gets the <a href="Canonical_commutation_relation" title="Canonical commutation relation">canonical commutation relation</a>, X⋅Y − Y⋅X = 1. Extending this relation by associativity and distributivity allows explicitly constructing the <a href="Weyl_algebra" title="Weyl algebra">Weyl algebra</a>. (<a href="#CITEREFLam2001">Lam 2001</a>, §1,ex1.9).

The skew-polynomial ring is defined similarly for a ring R and a ring <a href="Endomorphism" title="Endomorphism">endomorphism</a> f of R, by extending the multiplication from the relation X⋅r = f(r)⋅X to produce an associative multiplication that distributes over the standard addition. More generally, given a homomorphism F from the monoid N of the positive integers into the <a href="Endomorphism_ring" title="Endomorphism ring">endomorphism ring</a> of R, the formula Xn⋅r = F(n)(r)⋅Xn allows constructing a skew-polynomial ring. (<a href="#CITEREFLam2001">Lam 2001</a>, §1,ex 1.11) Skew polynomial rings are closely related to <a href="Crossed_product" title="Crossed product">crossed product</a> algebras.

<div class="mw-heading mw-heading3"><h3 id="Polynomial_rigs">Polynomial rigs</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Formal_power_series#On_a_semiring" title="Formal power series">Formal power series § On a semiring</a></div>

The definition of a polynomial ring can be generalised by relaxing the requirement that the algebraic structure R be a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> or a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> to the requirement that R only be a <a href="Semifield" title="Semifield">semifield</a> or <a href="Rig_(mathematics)" class="mw-redirect" title="Rig (mathematics)">rig</a>; the resulting polynomial structure/extension R[X] is a polynomial rig. For example, the set of all multivariate polynomials with <a href="Natural_number" title="Natural number">natural number</a> coefficients is a polynomial rig.

<ul><li><a href="Additive_polynomial" title="Additive polynomial">Additive polynomial</a></li>

<li><a href="Laurent_polynomial" title="Laurent polynomial">Laurent polynomial</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFHerstein1975">Herstein 1975</a>, p. 153

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Herstein, Hall p. 73

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, p. 97

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFHerstein1975">Herstein 1975</a>, p. 154

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, p. 100

</li>

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</style><cite id="CITEREFAntonBivensDavis2012" class="citation cs2">Anton, Howard; Bivens, Irl C.; Davis, Stephen (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=U2uv84cpJHQC&pg=RA1-PA31">Calculus Single Variable</a>, Wiley, p. 31, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780470647707</bdi></cite>.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFSendraWinklerPérez-Diaz2007" class="citation cs2">Sendra, J. Rafael; Winkler, Franz; Pérez-Diaz, Sonia (2007), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=puWxs7KG2D0C&pg=PA250">Rational Algebraic Curves: A Computer Algebra Approach</a>, Algorithms and Computation in Mathematics, vol. 22, Springer, p. 250, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783540737247</bdi></cite>.

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFEves1980" class="citation cs2"><a href="Howard_Eves" title="Howard Eves">Eves, Howard Whitley</a> (1980), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ayVxeUNbZRAC&pg=PA183">Elementary Matrix Theory</a>, Dover, p. 183, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780486150277</bdi></cite>.

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFHerstein1975">Herstein 1975</a>, pp. 155, 162

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="#CITEREFHerstein1975">Herstein 1975</a>, p. 162

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> Knapp, Anthony W. (2006), Basic Algebra, <a href="Birkh%C3%A4user" title="Birkhäuser">Birkhäuser</a>, p. 121.

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite id="CITEREFFröhlich,_A.Shepherson,_J._C.1955" class="citation cs2">Fröhlich, A.; Shepherson, J. C. (1955), "On the factorisation of polynomials in a finite number of steps", Mathematische Zeitschrift, 62 (1): 331–334, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01180640">10.1007/BF01180640</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5874">0025-5874</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119955899">119955899</a></cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFHall1969" class="citation cs2">Hall, F. M. (1969), "Section 3.6", <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoab0000hall_v1">An Introduction to Abstract Algebra</a>, vol. 2, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0521084849</bdi></cite></li>

<li><cite id="CITEREFHerstein1975" class="citation cs2">Herstein, I. N. (1975), "Section 3.9", <a rel="nofollow" class="external text" href="https://archive.org/details/topicsinalgebra00hers">Topics in Algebra</a>, Wiley, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0471010901</bdi>, <q>polynomial ring.</q></cite></li>

<li><cite id="CITEREFLam2001" class="citation cs2"><a href="Tsit_Yuen_Lam" title="Tsit Yuen Lam">Lam, Tsit-Yuen</a> (2001), A First Course in Noncommutative Rings, <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-95325-0</bdi></cite></li>

<li><cite id="CITEREFLang2002" class="citation cs2"><a href="Serge_Lang" title="Serge Lang">Lang, Serge</a> (2002), Algebra, <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 211 (Revised third ed.), New York: Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-95385-4</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1878556">1878556</a></cite></li>

<li><cite id="CITEREFOsborne2000" class="citation cs2">Osborne, M. Scott (2000), Basic homological algebra, Graduate Texts in Mathematics, vol. 196, <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-1278-2">10.1007/978-1-4612-1278-2</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-98934-1</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1757274">1757274</a></cite></li></ul>

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