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Simple extension

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">field theory</a>, a simple extension is a <a href="Field_extension" title="Field extension">field extension</a> that is generated by the <a href="Adjunction_(field_theory)" class="mw-redirect" title="Adjunction (field theory)">adjunction</a> of a single element, called a primitive element. Simple extensions are well understood and can be completely classified.

The <a href="Primitive_element_theorem" title="Primitive element theorem">primitive element theorem</a> provides a characterization of the <a href="Finite_extension" class="mw-redirect" title="Finite extension">finite</a> simple extensions.

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

A field extension L/K is called a simple extension if there exists an element θ in L with

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

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</math><img src="./5e72fea03262ffc24bd482ca7d560f7a2562e97e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.294ex; height:2.843ex;" alt="{\displaystyle L=K(\theta ).}" loading="lazy"></dd></dl>

This means that every element of L can be expressed as a <a href="Rational_fraction" class="mw-redirect" title="Rational fraction">rational fraction</a> in θ, with coefficients in K; that is, it is produced from θ and elements of K by the field operations +, −, •, / . Equivalently, L is the smallest field that contains both K and θ.

There are two different kinds of simple extensions (see <a href="#Structure_of_simple_extensions">§ Structure of simple extensions</a> below):

<ol><li>The element θ may be <a href="Transcendental_element" class="mw-redirect" title="Transcendental element">transcendental</a> over K, which means that it is not a <a href="Zero_of_a_function" title="Zero of a function">root</a> of any <a href="Polynomial" title="Polynomial">polynomial</a> with coefficients in K. In this case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\theta )}">

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

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</math><img src="./9a3a3b3be899442e25f8a41597be005cea095a9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.966ex; height:2.843ex;" alt="{\displaystyle K(\theta )}" loading="lazy"> is <a href="Ring_isomorphism" class="mw-redirect" title="Ring isomorphism">isomorphic</a> to the <a href="Field_of_rational_functions" class="mw-redirect" title="Field of rational functions">field of rational functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(X).}">

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

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</math><img src="./645ec14605ec0bf2c45f597fff13a9e87097fdf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.502ex; height:2.843ex;" alt="{\displaystyle K(X).}" loading="lazy"></li>

<li>Otherwise, θ is <a href="Algebraic_element" title="Algebraic element">algebraic</a> over K; that is, θ is a root of a polynomial over K. The <a href="Monic_polynomial" title="Monic polynomial">monic polynomial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(X)}">

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

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</math><img src="./f7425278aab7b6ceb8ddb54fa5ce71d2cf52d28f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.048ex; height:2.843ex;" alt="{\displaystyle p(X)}" loading="lazy"> of minimal <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> n, with θ as a root, is called the <a href="Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">minimal polynomial</a> of θ. Its degree equals the <a href="Degree_of_a_field_extension" title="Degree of a field extension">degree of the field extension</a>, that is, the <a href="Dimension_(vector_space)" title="Dimension (vector space)">dimension</a> of L viewed as a K-<a href="Vector_space" title="Vector space">vector space</a>. In this case, every element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\theta )}">

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

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</math><img src="./9a3a3b3be899442e25f8a41597be005cea095a9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.966ex; height:2.843ex;" alt="{\displaystyle K(\theta )}" loading="lazy"> can be uniquely expressed as a polynomial in θ of degree less than n, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\theta )}">

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

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</math><img src="./9a3a3b3be899442e25f8a41597be005cea095a9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.966ex; height:2.843ex;" alt="{\displaystyle K(\theta )}" loading="lazy"> is isomorphic to the <a href="Quotient_ring" title="Quotient ring">quotient ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/(p(X)).}">

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

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</math><img src="./33a822106d5c6d5af542992fca515226981a55b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.917ex; height:2.843ex;" alt="{\displaystyle K[X]/(p(X)).}" loading="lazy"></li></ol>

In both cases, the element θ is called a generating element or primitive element for the extension; one says also L is generated over K by θ.

For example, every <a href="Finite_field" title="Finite field">finite field</a> is a simple extension of the <a href="Prime_field" class="mw-redirect" title="Prime field">prime field</a> of the same <a href="Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a>. More precisely, if p is a prime number and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=p^{n},}">

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

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</math><img src="./70ecf365c0075d9e6a1694b9cc4bdc4ab33f640f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.203ex; height:2.676ex;" alt="{\displaystyle q=p^{n},}" loading="lazy"> the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\mathbb {F} _{q}}">

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<annotation encoding="application/x-tex">{\displaystyle L=\mathbb {F} _{q}}</annotation>

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</math><img src="./d9cd69c2e7b9f9c0e811f3c26370ab43c8554056.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.09ex; height:2.843ex;" alt="{\displaystyle L=\mathbb {F} _{q}}" loading="lazy"> of q elements is a simple extension of degree n of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {F} _{p}.}">

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<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {F} _{p}.}</annotation>

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</math><img src="./9d5f6d0e305c1118bd1491c5ee1d6ef2166906df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.291ex; height:2.843ex;" alt="{\displaystyle K=\mathbb {F} _{p}.}" loading="lazy"> In fact, L is generated as a field by any element θ that is a root of an <a href="Irreducible_polynomial" title="Irreducible polynomial">irreducible polynomial</a> of <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> n in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]}">

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

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</math><img src="./5bb4d802ca5718a14dc961af8692f35cdfad169b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.34ex; height:2.843ex;" alt="{\displaystyle K[X]}" loading="lazy">.

However, in the case of finite fields, the term primitive element is usually reserved for a stronger notion, an element γ that <a href="Generating_set_of_a_group" title="Generating set of a group">generates</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\times }=L-\{0\}}">

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<annotation encoding="application/x-tex">{\displaystyle L^{\times }=L-\{0\}}</annotation>

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</math><img src="./f353e1d78badc948c5ebd299628323814b1b9051.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.103ex; height:2.843ex;" alt="{\displaystyle L^{\times }=L-\{0\}}" loading="lazy"> as a <a href="Multiplicative_group" title="Multiplicative group">multiplicative group</a>, so that every nonzero element of L is a power of γ, i.e. is produced from γ using only the group operation • . To distinguish these meanings, one uses the term "generator" or field primitive element for the weaker meaning, reserving "primitive element" or group primitive element for the stronger meaning.<a href="#cite_note-1">[1]</a> (See <a href="Finite_field#Multiplicative_structure" title="Finite field">Finite field § Multiplicative structure</a> and <a href="Primitive_element_(finite_field)" title="Primitive element (finite field)">Primitive element (finite field)</a>).

<div class="mw-heading mw-heading2"><h2 id="Structure_of_simple_extensions">Structure of simple extensions</h2></div>

Let L be a simple extension of K generated by θ. For the <a href="Polynomial_ring" title="Polynomial ring">polynomial ring</a> K[X], one of its main properties is the unique <a href="Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varphi :K[X]&\rightarrow L\\f(X)&\mapsto f(\theta )\,.\end{aligned}}}</annotation>

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</math><img src="./8485c9914dd5813d748354154f144ba13e06d9eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.375ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\varphi :K[X]&\rightarrow L\\f(X)&\mapsto f(\theta )\,.\end{aligned}}}" loading="lazy"></dd></dl>

Two cases may occur:

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

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> is <a href="Injective" class="mw-redirect" title="Injective">injective</a>, it may be extended injectively to the <a href="Field_of_fractions" title="Field of fractions">field of fractions</a> K(X) of K[X]. Since L is generated by θ, this implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> is an isomorphism from K(X) onto L. This implies that every element of L is equal to an <a href="Irreducible_fraction" title="Irreducible fraction">irreducible fraction</a> of polynomials in θ, and that two such irreducible fractions are equal if and only if one may pass from one to the other by multiplying the numerator and the denominator by the same non zero element of K.</li>

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

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> is not injective, let p(X) be a generator of its <a href="Kernel_(algebra)#Ring_homomorphisms" title="Kernel (algebra)">kernel</a>, which is thus the <a href="Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">minimal polynomial</a> of θ. The <a href="Image_(mathematics)" title="Image (mathematics)">image</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> is a <a href="Subring" title="Subring">subring</a> of L, and thus an <a href="Integral_domain" title="Integral domain">integral domain</a>. This implies that p is an irreducible polynomial, and thus that the <a href="Quotient_ring" title="Quotient ring">quotient ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/\langle p(X)\rangle }">

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

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</math><img src="./7add84b5f3bc1e894117b9f63a8aa72cd1062c2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.27ex; height:2.843ex;" alt="{\displaystyle K[X]/\langle p(X)\rangle }" loading="lazy"> is a field. As L is generated by θ, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> is <a href="Surjective" class="mw-redirect" title="Surjective">surjective</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> induces an <a href="Isomorphism" title="Isomorphism">isomorphism</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/\langle p(X)\rangle }">

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

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</math><img src="./7add84b5f3bc1e894117b9f63a8aa72cd1062c2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.27ex; height:2.843ex;" alt="{\displaystyle K[X]/\langle p(X)\rangle }" loading="lazy"> onto L. This implies that every element of L is equal to a unique polynomial in θ of degree lower than the degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=\operatorname {deg} p(X)}">

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<annotation encoding="application/x-tex">{\displaystyle n=\operatorname {deg} p(X)}</annotation>

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</math><img src="./470150f8def0b8eb19ea2460227acd405f7057d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.326ex; height:2.843ex;" alt="{\displaystyle n=\operatorname {deg} p(X)}" loading="lazy">. That is, we have a K-basis of L given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,\theta ,\theta ^{2},\ldots ,\theta ^{n-1}}">

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<annotation encoding="application/x-tex">{\displaystyle 1,\theta ,\theta ^{2},\ldots ,\theta ^{n-1}}</annotation>

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</math><img src="./f2690ace4b62ddc534118e868e8165f7a95666b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.053ex; height:3.009ex;" alt="{\displaystyle 1,\theta ,\theta ^{2},\ldots ,\theta ^{n-1}}" loading="lazy">.</li></ol>

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

<ul><li>C / R generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =i={\sqrt {-1}}}">

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

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</math><img src="./aca00e0cb15688069dea55cc14298d7543f802c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.996ex; height:3.009ex;" alt="{\displaystyle \theta =i={\sqrt {-1}}}" loading="lazy">.</li>

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation>

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</math><img src="./b4afc1e27d418021bf10898eb44a7f5f315735ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}" loading="lazy">) / Q generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\sqrt {2}}}">

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<annotation encoding="application/x-tex">{\displaystyle \theta ={\sqrt {2}}}</annotation>

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</math><img src="./3a5b7f4e74326ca3a1375c08660f703dcf2c8b73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.287ex; height:3.009ex;" alt="{\displaystyle \theta ={\sqrt {2}}}" loading="lazy">.</li>

<li>Any <a href="Number_field" class="mw-redirect" title="Number field">number field</a> (i.e., a finite extension of Q) is a simple extension Q(θ) for some θ. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} ({\sqrt {3}},{\sqrt {7}})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} ({\sqrt {3}},{\sqrt {7}})}</annotation>

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</math><img src="./6b99e8a1f85263c63586a47505861c89da7e4450.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.048ex; height:3.176ex;" alt="{\displaystyle \mathbf {Q} ({\sqrt {3}},{\sqrt {7}})}" loading="lazy"> is generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\sqrt {3}}+{\sqrt {7}}}">

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<annotation encoding="application/x-tex">{\displaystyle \theta ={\sqrt {3}}+{\sqrt {7}}}</annotation>

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</math><img src="./8d986e362c7ca75eb77229a37968912598c49aec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.226ex; height:3.009ex;" alt="{\displaystyle \theta ={\sqrt {3}}+{\sqrt {7}}}" loading="lazy">.</li>

<li>F(X) / F, a field of rational functions, is generated by the formal variable X.</li></ul>

<ul><li><a href="Companion_matrix#Multiplication_map_on_a_simple_field_extension" title="Companion matrix">Companion matrix</a> for the multiplication map on a simple field extension</li></ul>

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

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> (<a href="#CITEREFRoman1995">Roman 1995</a>)

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

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</style><cite id="CITEREFRoman1995" class="citation book cs1"><a href="Steven_Roman" title="Steven Roman">Roman, Steven</a> (1995). Field Theory. <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>. Vol. 158. New York: <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>0-387-94408-7</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0816.12001">0816.12001</a>.</cite></li></ul></div><div><div class="zim-footer">

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