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Algebraic integer

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the ring of complex numbers integral over <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="./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">. For the general notion of algebraic integer, see <a href="Integrality" class="mw-redirect" title="Integrality">Integrality</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Algebraic_element" title="Algebraic element">algebraic element</a> or <a href="Algebraic_number" title="Algebraic number">algebraic number</a>.</div>

In <a href="Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>, an algebraic integer is a <a href="Complex_number" title="Complex number">complex number</a> that is <a href="Integral_element" title="Integral element">integral</a> over the <a href="Integer#Algebraic_properties" title="Integer">integers</a>. That is, an algebraic integer is a complex <a href="Root_of_a_polynomial" class="mw-redirect" title="Root of a polynomial">root</a> of some <a href="Monic_polynomial" title="Monic polynomial">monic polynomial</a> (a <a href="Polynomial" title="Polynomial">polynomial</a> whose <a href="Leading_coefficient" class="mw-redirect" title="Leading coefficient">leading coefficient</a> is 1) whose coefficients are integers. The set of all algebraic integers A is closed under addition, subtraction and multiplication and therefore is a <a href="Commutative_ring" title="Commutative ring">commutative</a> <a href="Subring" title="Subring">subring</a> of the complex numbers.

The <a href="Ring_of_integers" title="Ring of integers">ring of integers</a> of a <a href="Number_field" class="mw-redirect" title="Number field">number field</a> K, denoted by OK, is the <a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> of K and A: it can also be characterized as the maximal <a href="Order_(ring_theory)" title="Order (ring theory)">order</a> of the <a href="Field_(mathematics)" title="Field (mathematics)">field</a> K. Each algebraic integer belongs to the ring of integers of some number field. A number α is an algebraic integer <a href="If_and_only_if" title="If and only if">if and only if</a> the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [\alpha ]}">

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

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</math><img src="./28420682be585a0ee588c7eb13e7971e01cb1283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.332ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [\alpha ]}" loading="lazy"> is <a href="Finitely_generated_abelian_group" title="Finitely generated abelian group">finitely generated</a> as an <a href="Abelian_group" title="Abelian group">abelian group</a>, which is to say, as a <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="./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">-<a href="Module_(mathematics)" title="Module (mathematics)">module</a>.

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

The following are equivalent definitions of an algebraic integer. Let K be a <a href="Number_field" class="mw-redirect" title="Number field">number field</a> (i.e., a <a href="Finite_extension" class="mw-redirect" title="Finite extension">finite extension</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

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

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</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy">, the field of <a href="Rational_number" title="Rational number">rational numbers</a>), in other words, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {Q} (\theta )}">

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

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</math><img src="./2ea639a5a826d082a3eef02bf04dad8eb62fac51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.872ex; height:2.843ex;" alt="{\displaystyle K=\mathbb {Q} (\theta )}" loading="lazy"> for some <a href="Algebraic_number" title="Algebraic number">algebraic number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \in \mathbb {C} }">

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

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</math><img src="./db50602b6cc4677d40a83e8489738c3feff0b16e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.609ex; height:2.176ex;" alt="{\displaystyle \theta \in \mathbb {C} }" loading="lazy"> by the <a href="Primitive_element_theorem" title="Primitive element theorem">primitive element theorem</a>.

<ul><li>α ∈ K is an algebraic integer if there exists a monic polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\in \mathbb {Z} [x]}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)\in \mathbb {Z} [x]}</annotation>

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</math><img src="./cb8266c32dfc2904dc4331ea8906ad341a05ce1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.432ex; height:2.843ex;" alt="{\displaystyle f(x)\in \mathbb {Z} [x]}" loading="lazy"> such that f(α) = 0.</li>

<li>α ∈ K is an algebraic integer if the <a href="Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">minimal</a> monic polynomial of α over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

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

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</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> is in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [x]}">

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

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</math><img src="./0d4da3ac703cc7721ebba91a53f6752de7157124.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.174ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [x]}" loading="lazy">.</li>

<li>α ∈ K is an algebraic integer if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [\alpha ]}">

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

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</math><img src="./28420682be585a0ee588c7eb13e7971e01cb1283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.332ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [\alpha ]}" loading="lazy"> is a finitely generated <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="./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">-module.</li>

<li>α ∈ K is an algebraic integer if there exists a non-zero finitely generated <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="./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">-<a href="Submodule" class="mw-redirect" title="Submodule">submodule</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\subset \mathbb {C} }">

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

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</math><img src="./067e5aa6a8937fe898a8f4289e7f29b591487792.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.219ex; height:2.176ex;" alt="{\displaystyle M\subset \mathbb {C} }" loading="lazy"> such that αM ⊆ M.</li></ul>

Algebraic integers are a special case of <a href="Integral_element" title="Integral element">integral elements</a> of a ring extension. In particular, an algebraic integer is an integral element of a finite extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/\mathbb {Q} }">

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

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</math><img src="./ce6782a2935122dafec2947d1d7ba168be4399ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.037ex; height:2.843ex;" alt="{\displaystyle K/\mathbb {Q} }" loading="lazy">.

Note that if P(x) is a <a href="Primitive_polynomial_(ring_theory)" class="mw-redirect" title="Primitive polynomial (ring theory)">primitive polynomial</a> that has integer coefficients but is not monic, and P is <a href="Irreducible_polynomial" title="Irreducible polynomial">irreducible</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

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

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</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy">, then none of the roots of P are algebraic integers (but are <a href="Algebraic_number" title="Algebraic number">algebraic numbers</a>). Here primitive is used in the sense that the <a href="Highest_common_factor" class="mw-redirect" title="Highest common factor">highest common factor</a> of the coefficients of P is 1, which is weaker than requiring the coefficients to be pairwise relatively prime.

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

<ul><li>The only algebraic integers that are found in the set of rational numbers are the integers. In other words, the intersection of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

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

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</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> and A is exactly <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="./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">. The rational number <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠a/b⁠ is not an algebraic integer unless b <a href="Divisor" title="Divisor">divides</a> a. The leading coefficient of the polynomial bx − a is the integer b.</li>

<li>The <a href="Square_root" title="Square root">square root</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}}">

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

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</math><img src="./2a2994734eae382ce30100fb17b9447fd8e99f81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.331ex; height:3.009ex;" alt="{\displaystyle {\sqrt {n}}}" loading="lazy"> of a nonnegative integer n is an algebraic integer, but is <a href="Irrational_number" title="Irrational number">irrational</a> unless n is a <a href="Square_number" title="Square number">perfect square</a>.</li>

<li>If d is a <a href="Square-free_integer" title="Square-free integer">square-free integer</a> then the <a href="Field_extension" title="Field extension">extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,)}">

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<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,)}</annotation>

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</math><img src="./f8d581189704afb02a5ec511112bac58b475ad2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.321ex; height:3.176ex;" alt="{\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,)}" loading="lazy"> is a <a href="Quadratic_field_extension" class="mw-redirect" title="Quadratic field extension">quadratic field</a> of rational numbers. The ring of algebraic integers OK contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {d}}}">

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

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</math><img src="./67bee2dadb035f786486ce4dc3bc46f30f66cb7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.152ex; height:3.009ex;" alt="{\displaystyle {\sqrt {d}}}" loading="lazy"> since this is a root of the monic polynomial x2 − d. Moreover, if d ≡ 1 <a href="Modular_arithmetic" title="Modular arithmetic">mod</a> 4, then the element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{2}}(1+{\sqrt {d}}\,)}">

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<annotation encoding="application/x-tex">{\textstyle {\frac {1}{2}}(1+{\sqrt {d}}\,)}</annotation>

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</math><img src="./ed224dbe2364da14b9506e2fca4078efebba253c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.009ex; height:3.509ex;" alt="{\textstyle {\frac {1}{2}}(1+{\sqrt {d}}\,)}" loading="lazy"> is also an algebraic integer. It satisfies the polynomial x2 − x + ⁠1/4⁠(1 − d) where the <a href="Constant_term" title="Constant term">constant term</a> ⁠1/4⁠(1 − d) is an integer. The full ring of integers is generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {d}}}">

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

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</math><img src="./67bee2dadb035f786486ce4dc3bc46f30f66cb7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.152ex; height:3.009ex;" alt="{\displaystyle {\sqrt {d}}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{2}}(1+{\sqrt {d}}\,)}">

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<annotation encoding="application/x-tex">{\textstyle {\frac {1}{2}}(1+{\sqrt {d}}\,)}</annotation>

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</math><img src="./ed224dbe2364da14b9506e2fca4078efebba253c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.009ex; height:3.509ex;" alt="{\textstyle {\frac {1}{2}}(1+{\sqrt {d}}\,)}" loading="lazy"> respectively. See <a href="Quadratic_integer" title="Quadratic integer">Quadratic integer</a> for more.</li>

<li>The ring of integers of the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=\mathbb {Q} [\alpha ]}">

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<annotation encoding="application/x-tex">{\displaystyle F=\mathbb {Q} [\alpha ]}</annotation>

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</math><img src="./18fe97465addcc1c7d3f2a3a68e369894145b652.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.429ex; height:2.843ex;" alt="{\displaystyle F=\mathbb {Q} [\alpha ]}" loading="lazy">, α = 3√m, has the following <a href="Integral_basis" class="mw-redirect" title="Integral basis">integral basis</a>, writing m = hk2 for two <a href="Square-free_integer" title="Square-free integer">square-free</a> <a href="Coprime" class="mw-redirect" title="Coprime">coprime</a> integers h and k:<a href="#cite_note-1">[1]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}1,\alpha ,{\dfrac {\alpha ^{2}\pm k^{2}\alpha +k^{2}}{3k}}&m\equiv \pm 1{\bmod {9}}\\1,\alpha ,{\dfrac {\alpha ^{2}}{k}}&{\text{otherwise}}\end{cases}}}">

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<mtext>otherwise</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}1,\alpha ,{\dfrac {\alpha ^{2}\pm k^{2}\alpha +k^{2}}{3k}}&m\equiv \pm 1{\bmod {9}}\\1,\alpha ,{\dfrac {\alpha ^{2}}{k}}&{\text{otherwise}}\end{cases}}}</annotation>

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</math></li>

<li>If ζn is a <a href="Primitive_root_of_unity" class="mw-redirect" title="Primitive root of unity">primitive</a> nth <a href="Root_of_unity" title="Root of unity">root of unity</a>, then the ring of integers of the <a href="Cyclotomic_field" title="Cyclotomic field">cyclotomic field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} (\zeta _{n})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (\zeta _{n})}</annotation>

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</math><img src="./1cb4399e6836cf4788d5f0915207127791b68aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.854ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} (\zeta _{n})}" loading="lazy"> is precisely <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [\zeta _{n}]}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [\zeta _{n}]}</annotation>

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</math><img src="./77d56918ae77f0c6d51f8404ce9240aacca9780e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.081ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [\zeta _{n}]}" loading="lazy">.</li>

<li>If α is an algebraic integer then β = n√α is another algebraic integer. A polynomial for β is obtained by substituting xn in the polynomial for α.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Finite_generation_of_ring_extension">Finite generation of ring extension</h2></div>

For any α, the <a href="Subring#Ring_extensions" title="Subring">ring extension</a> (in the sense that is equivalent to <a href="Field_extension" title="Field extension">field extension</a>) of the integers by α, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [\alpha ]\equiv \left\{\sum _{i=0}^{n}\alpha ^{i}z_{i}|z_{i}\in \mathbb {Z} ,n\in \mathbb {Z} \right\}}">

</mrow>

<mrow>

<mrow>

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</munderover>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

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</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [\alpha ]\equiv \left\{\sum _{i=0}^{n}\alpha ^{i}z_{i}|z_{i}\in \mathbb {Z} ,n\in \mathbb {Z} \right\}}</annotation>

</semantics>

</math><img src="./bd0afabb0e882508f14afd72027f19ccdfad4358.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.825ex; height:7.509ex;" alt="{\displaystyle \mathbb {Z} [\alpha ]\equiv \left\{\sum _{i=0}^{n}\alpha ^{i}z_{i}|z_{i}\in \mathbb {Z} ,n\in \mathbb {Z} \right\}}" loading="lazy">, is <a href="Finitely_generated_abelian_group" title="Finitely generated abelian group">finitely generated</a> if and only if α is an algebraic integer.

The proof is analogous to that of the <a href="Algebraic_number#Degree_of_simple_extensions_of_the_rationals_as_a_criterion_to_algebraicity" title="Algebraic number">corresponding fact</a> regarding <a href="Algebraic_number" title="Algebraic number">algebraic numbers</a>, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

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</mrow>

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

</semantics>

</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> there replaced by <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"> here, and the notion of <a href="Degree_of_a_field_extension" title="Degree of a field extension">field extension degree</a> replaced by finite generation (using the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<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 finitely generated itself); the only required change is that only non-negative powers of α are involved in the proof.

The analogy is possible because both algebraic integers and algebraic numbers are defined as roots of monic polynomials over either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

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

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</mrow>

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

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</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy">, respectively.

The sum, difference and product of two algebraic integers is an algebraic integer. In general their quotient is not. Thus the algebraic integers form a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>.

This can be shown analogously to <a href="Algebraic_number#Field" title="Algebraic number">the corresponding proof</a> for <a href="Algebraic_number" title="Algebraic number">algebraic numbers</a>, using the integers <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"> instead of the rationals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

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

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</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy">.

One may also construct explicitly the monic polynomial involved, which is generally of higher <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> than those of the original algebraic integers, by taking <a href="Resultant" title="Resultant">resultants</a> and factoring. For example, if x2 − x − 1 = 0, y3 − y − 1 = 0 and z = xy, then eliminating x and y from z − xy = 0 and the polynomials satisfied by x and y using the resultant gives z6 − 3z4 − 4z3 + z2 + z − 1 = 0, which is irreducible, and is the monic equation satisfied by the product. (To see that the xy is a root of the x-resultant of z − xy and x2 − x − 1, one might use the fact that the resultant is contained in the <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> generated by its two input polynomials.)

<div class="mw-heading mw-heading3"><h3 id="Integral_closure">Integral closure</h3></div>

Every root of a monic polynomial whose coefficients are algebraic integers is itself an algebraic integer. In other words, the algebraic integers form a ring that is <a href="Integrally_closed_domain" title="Integrally closed domain">integrally closed</a> in any of its extensions.

Again, the proof is analogous to <a href="Algebraic_number#Algebraic_closure" title="Algebraic number">the corresponding proof</a> for <a href="Algebraic_number" title="Algebraic number">algebraic numbers</a> being <a href="Algebraically_closed_field" title="Algebraically closed field">algebraically closed</a>.

<div class="mw-heading mw-heading2"><h2 id="Additional_facts">Additional facts</h2></div>

<ul><li>Any number constructible out of the integers with roots, addition, and multiplication is an algebraic integer; but not all algebraic integers are so constructible: in a naïve sense, most roots of irreducible <a href="Quintic" class="mw-redirect" title="Quintic">quintics</a> are not. This is the <a href="Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a>.</li>

<li>The ring of algebraic integers is a <a href="B%C3%A9zout_domain" title="Bézout domain">Bézout domain</a>, as a consequence of the <a href="Principal_ideal_theorem" title="Principal ideal theorem">principal ideal theorem</a>.</li>

<li>If the monic polynomial associated with an algebraic integer has constant term 1 or −1, then the <a href="Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> of that algebraic integer is also an algebraic integer, and each is a <a href="Unit_(ring_theory)" title="Unit (ring theory)">unit</a>, an element of the <a href="Group_of_units" class="mw-redirect" title="Group of units">group of units</a> of the ring of algebraic integers.</li>

<li>If x is an algebraic number then anx is an algebraic integer, where x satisfies a polynomial p(x) with integer coefficients and where anxn is the highest-degree term of p(x). The value y = anx is an algebraic integer because it is a root of q(y) = an − 1 n p(y /an), where q(y) is a monic polynomial with integer coefficients.</li>

<li>If x is an algebraic number then it can be written as the ratio of an algebraic integer to a non-zero algebraic integer. In fact, the denominator can always be chosen to be a positive integer. The ratio is |an|x / |an|, where x satisfies a polynomial p(x) with integer coefficients and where anxn is the highest-degree term of p(x).</li>

<li>The only rational algebraic integers are the integers. That is, if x is an algebraic integer and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {Q} }">

</mrow>

</mstyle>

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

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</math><img src="./eae13dca0f57812b8bb2eae6a3bb0a00321ba46a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.978ex; height:2.509ex;" alt="{\displaystyle x\in \mathbb {Q} }" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {Z} }">

</mrow>

</mstyle>

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

</semantics>

</math><img src="./fff96cba8be0fa52dd0b472c9f40e7d6a3cb75db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.721ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {Z} }" loading="lazy">. This is a direct result of the <a href="Rational_root_theorem" title="Rational root theorem">rational root theorem</a> for the case of a monic polynomial.</li></ul>

<ul><li><a href="Gaussian_integer" title="Gaussian integer">Gaussian integer</a></li>

<li><a href="Eisenstein_integer" title="Eisenstein integer">Eisenstein integer</a></li>

<li><a href="Root_of_unity" title="Root of unity">Root of unity</a></li>

<li><a href="Dirichlet's_unit_theorem" title="Dirichlet's unit theorem">Dirichlet's unit theorem</a></li>

<li><a href="Fundamental_unit_(number_theory)" title="Fundamental unit (number theory)">Fundamental units</a></li></ul>

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

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</style><cite id="CITEREFMarcus1977" class="citation book cs1">Marcus, Daniel A. (1977). Number Fields (3rd ed.). Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. ch. 2, p. 38 and ex. 41. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90279-1</bdi>.</cite>

</li>

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<ul><li><cite id="CITEREFStein" class="citation book cs1"><a href="William_A._Stein" title="William A. Stein">Stein, William</a>. <a rel="nofollow" class="external text" href="https://wstein.org/books/ant/ant.pdf">Algebraic Number Theory: A Computational Approach</a> (PDF). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131102070632/http://wstein.org/books/ant/ant.pdf">Archived</a> (PDF) from the original on November 2, 2013.</cite></li></ul>

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</style><div id="Algebraic_numbers198" style="font-size:114%;margin:0 4em"><a href="Algebraic_number" title="Algebraic number">Algebraic numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul>

<li><a href="Chebyshev_nodes" title="Chebyshev nodes">Chebyshev nodes</a></li>

<li><a href="Constructible_number" title="Constructible number">Constructible number</a></li>

<li><a href="Look-and-say_sequence" title="Look-and-say sequence">Conway's constant</a></li>

<li><a href="Cyclotomic_field" title="Cyclotomic field">Cyclotomic field</a></li>

<li><a href="Doubling_the_cube" title="Doubling the cube">Doubling the cube</a></li>

<li><a href="Eisenstein_integer" title="Eisenstein integer">Eisenstein integer</a></li>

<li><a href="Gaussian_integer" title="Gaussian integer">Gaussian integer</a></li>

<li><a href="Golden_ratio" title="Golden ratio">Golden ratio (φ)</a></li>

<li><a href="Perron_number" title="Perron number">Perron number</a></li>

<li><a href="Pisot%E2%80%93Vijayaraghavan_number" title="Pisot–Vijayaraghavan number">Pisot–Vijayaraghavan number</a></li>

<li><a href="Plastic_ratio" title="Plastic ratio">Plastic ratio (ρ)</a></li>

<li><a href="Quadratic_irrational_number" title="Quadratic irrational number">Quadratic irrational number</a></li>

<li><a href="Rational_number" title="Rational number">Rational number</a></li>

<li><a href="Root_of_unity" title="Root of unity">Root of unity</a></li>

<li><a href="Salem_number" title="Salem number">Salem number</a></li>

<li><a href="Silver_ratio" title="Silver ratio">Silver ratio (σ)</a></li>

<li><a href="Square_root_of_2" title="Square root of 2">Square root of 2</a></li>

<li><a href="Square_root_of_3" title="Square root of 3">Square root of 3</a></li>

<li><a href="Square_root_of_5" title="Square root of 5">Square root of 5</a></li>

<li><a href="Square_root_of_6" title="Square root of 6">Square root of 6</a></li>

<li><a href="Square_root_of_7" title="Square root of 7">Square root of 7</a></li>

<li><a href="Square_root_of_10" title="Square root of 10">Square root of 10</a></li>

<li><a href="Supergolden_ratio" title="Supergolden ratio">Supergolden ratio (ψ)</a></li>

<li><a href="Supersilver_ratio" title="Supersilver ratio">Supersilver ratio (ς)</a></li>

<li><a href="Twelfth_root_of_2" class="mw-redirect" title="Twelfth root of 2">Twelfth root of 2</a></li></ul>

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