Algebraic integer

<h2 id="definitions">Definitions</h2>
The following are equivalent definitions of an algebraic integer. Let K be a <a href="/facts/Number_field/e0K0TXSh">number field</a> (i.e., a <a href="/facts/Finite_extension/hY5LgHYk">finite extension</a> of 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
, the field of <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>), in other words, 
 
 
 
 K
 =
 
 Q
 
 (
 θ
 )
 
 
 {\displaystyle K=\mathbb {Q} (\theta )}
 
 for some <a href="/facts/Algebraic_number/MkH1PzTK">algebraic number</a> 
 
 
 
 θ
 ∈
 
 C
 
 
 
 {\displaystyle \theta \in \mathbb {C} }
 
 by the <a href="/facts/Primitive_element_theorem/u5DvBoTl">primitive element theorem</a>.

<ul><li>α ∈ K is an algebraic integer if there exists a monic polynomial 
 
 
 
 f
 (
 x
 )
 ∈
 
 Z
 
 [
 x
 ]
 
 
 {\displaystyle f(x)\in \mathbb {Z} [x]}
 
 such that f(α) = 0.</li>
<li>α ∈ K is an algebraic integer if the <a href="/facts/Minimal_polynomial_(field_theory)/dJv9Q0Kh">minimal</a> monic polynomial of α over 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
 is in 
 
 
 
 
 Z
 
 [
 x
 ]
 
 
 {\displaystyle \mathbb {Z} [x]}
 
.</li>
<li>α ∈ K is an algebraic integer if 
 
 
 
 
 Z
 
 [
 α
 ]
 
 
 {\displaystyle \mathbb {Z} [\alpha ]}
 
 is a finitely generated 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
-module.</li>
<li>α ∈ K is an algebraic integer if there exists a non-zero finitely generated 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
-<a href="/facts/Submodule/jP0LIhFL">submodule</a> 
 
 
 
 M
 ⊂
 
 C
 
 
 
 {\displaystyle M\subset \mathbb {C} }
 
 such that αM ⊆ M.</li></ul>
Algebraic integers are a special case of <a href="/facts/Integral_element/55dAuE39">integral elements</a> of a ring extension. In particular, an algebraic integer is an integral element of a finite extension 
 
 
 
 K
 
 /
 
 
 Q
 
 
 
 {\displaystyle K/\mathbb {Q} }
 
.

<h2 id="examples">Examples</h2>
<ul><li>The only algebraic integers that are found in the set of rational numbers are the integers. In other words, the intersection of 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
 and A is exactly 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
. The rational number ⁠a/b⁠ is not an algebraic integer unless b <a href="/facts/Divisor/DKNa2GvC">divides</a> a. The leading coefficient of the polynomial bx − a is the integer b.</li>
<li>The <a href="/facts/Square_root/AfrzfBdQ">square root</a> 
 
 
 
 
 
 n
 
 
 
 
 {\displaystyle {\sqrt {n}}}
 
 of a nonnegative integer n is an algebraic integer, but is <a href="/facts/Irrational_number/Psv19TET">irrational</a> unless n is a <a href="/facts/Square_number/jfakASPK">perfect square</a>.</li>
<li>If d is a <a href="/facts/Square-free_integer/R3qR9hP1">square-free integer</a> then the <a href="/facts/Field_extension/L7yIDtyK">extension</a> 
 
 
 
 K
 =
 
 Q
 
 (
 
 
 d
 
 
 
 )
 
 
 {\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,)}
 
 is a <a href="/facts/Quadratic_field_extension/dUnSDN4e">quadratic field</a> of rational numbers. The ring of algebraic integers OK contains 
 
 
 
 
 
 d
 
 
 
 
 {\displaystyle {\sqrt {d}}}
 
 since this is a root of the monic polynomial x2 − d. Moreover, if d ≡ 1 <a href="/facts/Modular_arithmetic/0wtRa5dU">mod</a> 4, then the element 
 
 
 
 
 
 1
 2
 
 
 (
 1
 +
 
 
 d
 
 
 
 )
 
 
 {\textstyle {\frac {1}{2}}(1+{\sqrt {d}}\,)}
 
 is also an algebraic integer. It satisfies the polynomial x2 − x + ⁠1/4⁠(1 − d) where the <a href="/facts/Constant_term/KRFu0aum">constant term</a> ⁠1/4⁠(1 − d) is an integer. The full ring of integers is generated by 
 
 
 
 
 
 d
 
 
 
 
 {\displaystyle {\sqrt {d}}}
 
 or 
 
 
 
 
 
 1
 2
 
 
 (
 1
 +
 
 
 d
 
 
 
 )
 
 
 {\textstyle {\frac {1}{2}}(1+{\sqrt {d}}\,)}
 
 respectively. See <a href="/facts/Quadratic_integer/uGsLh5kL">Quadratic integer</a> for more.</li>
<li>The ring of integers of the field 
 
 
 
 F
 =
 
 Q
 
 [
 α
 ]
 
 
 {\displaystyle F=\mathbb {Q} [\alpha ]}
 
, α = 3√m, has the following <a href="/facts/Integral_basis/oaSRGmpY">integral basis</a>, writing m = hk2 for two <a href="/facts/Square-free_integer/R3qR9hP1">square-free</a> <a href="/facts/Coprime/bnCxCXxs">coprime</a> integers h and k:<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> 
 
 
 
 
 
 {
 
 
 
 1
 ,
 α
 ,
 
 
 
 
 
 α
 
 2
 
 
 ±
 
 k
 
 2
 
 
 α
 +
 
 k
 
 2
 
 
 
 
 3
 k
 
 
 
 
 
 
 m
 ≡
 ±
 1
 
 mod
 
 9
 
 
 
 
 
 
 1
 ,
 α
 ,
 
 
 
 
 α
 
 2
 
 
 k
 
 
 
 
 
 
 otherwise
 
 
 
 
 
 
 
 
 
 {\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}}}
 
</li>
<li>If ζn is a <a href="/facts/Primitive_root_of_unity/EqH0ho1Y">primitive</a> nth <a href="/facts/Root_of_unity/EqH0ho1Y">root of unity</a>, then the ring of integers of the <a href="/facts/Cyclotomic_field/cGxMMEoR">cyclotomic field</a> 
 
 
 
 
 Q
 
 (
 
 ζ
 
 n
 
 
 )
 
 
 {\displaystyle \mathbb {Q} (\zeta _{n})}
 
 is precisely 
 
 
 
 
 Z
 
 [
 
 ζ
 
 n
 
 
 ]
 
 
 {\displaystyle \mathbb {Z} [\zeta _{n}]}
 
.</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>
<h2 id="non-example">Non-example</h2>
<ul><li>If P(x) is a <a href="/facts/Primitive_polynomial_(ring_theory)/JjnP4JCb">primitive polynomial</a> that has integer coefficients but is not monic, and P is <a href="/facts/Irreducible_polynomial/PTeuKm4W">irreducible</a> over 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
, then none of the roots of P are algebraic integers (but are <a href="/facts/Algebraic_number/MkH1PzTK">algebraic numbers</a>). Here primitive is used in the sense that the <a href="/facts/Highest_common_factor/kNFvBuKl">highest common factor</a> of the coefficients of P is 1, which is weaker than requiring the coefficients to be pairwise relatively prime.</li></ul>
<h2 id="finite-generation-of-ring-extension">Finite generation of ring extension</h2>
For any α, the <a href="/facts/Subring/M0perybt">ring extension</a> (in the sense that is equivalent to <a href="/facts/Field_extension/L7yIDtyK">field extension</a>) of the integers by α, denoted by 
 
 
 
 
 Z
 
 (
 α
 )
 ≡
 {
 
 ∑
 
 i
 =
 0
 
 
 n
 
 
 
 α
 
 i
 
 
 
 z
 
 i
 
 
 
 |
 
 
 z
 
 i
 
 
 ∈
 
 Z
 
 ,
 n
 ∈
 
 Z
 
 }
 
 
 {\displaystyle \mathbb {Z} (\alpha )\equiv \{\sum _{i=0}^{n}\alpha ^{i}z_{i}|z_{i}\in \mathbb {Z} ,n\in \mathbb {Z} \}}
 
, is <a href="/facts/Finitely_generated_abelian_group/dwtl7mp4">finitely generated</a> if and only if α is an algebraic integer.
The proof is analogous to that of the <a href="/facts/Algebraic_number/MkH1PzTK">corresponding fact</a> regarding <a href="/facts/Algebraic_number/MkH1PzTK">algebraic numbers</a>, with 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
 there replaced by 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 here, and the notion of <a href="/facts/Degree_of_a_field_extension/hY5LgHYk">field extension degree</a> replaced by finite generation (using the fact that 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 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 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 or 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
, respectively.

<h2 id="ring">Ring</h2>
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="/facts/Ring_(mathematics)/JnCUXz63">ring</a>.
This can be shown analogously to <a href="/facts/Algebraic_number/MkH1PzTK">the corresponding proof</a> for <a href="/facts/Algebraic_number/MkH1PzTK">algebraic numbers</a>, using the integers 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 instead of the rationals 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
.
One may also construct explicitly the monic polynomial involved, which is generally of higher <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> than those of the original algebraic integers, by taking <a href="/facts/Resultant/A5g2MdhT">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="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> generated by its two input polynomials.)

<h3>Integral closure</h3>
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="/facts/Integrally_closed_domain/L3kgG2q6">integrally closed</a> in any of its extensions.
Again, the proof is analogous to <a href="/facts/Algebraic_number/MkH1PzTK">the corresponding proof</a> for <a href="/facts/Algebraic_number/MkH1PzTK">algebraic numbers</a> being <a href="/facts/Algebraically_closed_field/HuVQClok">algebraically closed</a>.

<h2 id="additional-facts">Additional facts</h2>
<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="/facts/Quintic/oeNDGj9l">quintics</a> are not. This is the <a href="/facts/Abel%25E2%2580%2593Ruffini_theorem/tEFaRGvW">Abel–Ruffini theorem</a>.</li>
<li>The ring of algebraic integers is a <a href="/facts/B%25C3%25A9zout_domain/GXD2DJ0J">Bézout domain</a>, as a consequence of the <a href="/facts/Principal_ideal_theorem/usuL4GyJ">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="/facts/Multiplicative_inverse/yczdYyCw">reciprocal</a> of that algebraic integer is also an algebraic integer, and each is a <a href="/facts/Unit_(ring_theory)/GJSBu8Y7">unit</a>, an element of the <a href="/facts/Group_of_units/GJSBu8Y7">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 − 1n 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. Thus, if α is an algebraic integers and 
 
 
 
 α
 ∈
 
 Q
 
 
 
 {\displaystyle \alpha \in \mathbb {Q} }
 
, then 
 
 
 
 α
 ∈
 
 Z
 
 
 
 {\displaystyle \alpha \in \mathbb {Z} }
 
. This is a direct result of the <a href="/facts/Rational_root_theorem/U97kHx9n">rational root theorem</a> for the case of a monic polynomial.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Integral_element/55dAuE39">Integral element</a></li>
<li><a href="/facts/Gaussian_integer/zj82AhtL">Gaussian integer</a></li>
<li><a href="/facts/Eisenstein_integer/UGQmFI8c">Eisenstein integer</a></li>
<li><a href="/facts/Root_of_unity/EqH0ho1Y">Root of unity</a></li>
<li><a href="/facts/Dirichlet%2527s_unit_theorem/IQDk3QAR">Dirichlet's unit theorem</a></li>
<li><a href="/facts/Fundamental_unit_(number_theory)/T3dJERRQ">Fundamental units</a></li></ul>

<ul><li><a href="/facts/William_A._Stein/tBqUDE0P">Stein, William</a>. <a href="https://wstein.org/books/ant/ant.pdf">Algebraic Number Theory: A Computational Approach</a> (PDF). <a href="https://web.archive.org/web/20131102070632/http://wstein.org/books/ant/ant.pdf">Archived</a> (PDF) from the original on November 2, 2013.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Marcus, Daniel A. (1977). Number Fields (3rd ed.). Berlin, New York: Springer-Verlag. ch. 2, p. 38 and ex. 41. ISBN 978-0-387-90279-1. <a href="978-0-387-90279-1" target="_blank">978-0-387-90279-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Algebraic integer open-in-new

Algebraic integer