Editing Algebraic integer

{{Short description|Complex number that solves a monic polynomial with integer coefficients
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{{about|the ring of complex numbers integral over <math>\mathbb{Z}</math>|the general notion of algebraic integer|Integrality}}
{{Distinguish|algebraic element|algebraic number}}
{{use mdy dates|date=September 2021}}
{{Use American English|date=January 2019}}
In [[algebraic number theory]], an '''algebraic integer''' is a [[complex number]] that is [[Integral element|integral]] over the [[Integer#Algebraic properties|integers]]. That is, an algebraic integer is a complex [[root of a polynomial|root]] of some [[monic polynomial]] (a [[polynomial]] whose [[leading coefficient]] is 1) whose coefficients are integers. The set of all algebraic integers {{mvar|A}} is closed under addition, subtraction and multiplication and therefore is a [[commutative ring|commutative]] [[subring]] of the complex numbers.

The [[ring of integers]] of a [[number field]] {{mvar|K}}, denoted by {{math|{{mathcal|O}}<sub>''K''</sub>}}, is the [[intersection (set theory)|intersection]] of {{mvar|K}} and {{mvar|A}}: it can also be characterised as the maximal [[Order (ring theory)|order]] of the [[field (mathematics)|field]] {{mvar|K}}. Each algebraic integer belongs to the ring of integers of some number field. A number {{mvar|α}} is an algebraic integer [[if and only if]] the ring <math>\mathbb{Z}[\alpha]</math> is [[finitely generated abelian group|finitely generated]] as an [[abelian group]], which is to say, as a <math>\mathbb{Z}</math>-[[module (mathematics)|module]].

==Definitions==

The following are equivalent definitions of an algebraic integer. Let {{mvar|K}} be a [[number field]] (i.e., a [[finite extension]] of <math>\mathbb{Q}</math>, the field of [[rational number]]s), in other words, <math>K = \Q(\theta)</math> for some [[algebraic number]] <math>\theta \in \Complex</math> by the [[primitive element theorem]].

* {{math|''α'' ∈ ''K''}} is an algebraic integer if there exists a monic polynomial <math>f(x) \in \Z[x]</math> such that {{math|1=''f''(''α'') = 0}}.
* {{math|''α'' ∈ ''K''}} is an algebraic integer if the [[minimal polynomial (field theory)|minimal]] monic polynomial of {{mvar|α}} over <math>\mathbb{Q}</math> is in <math>\Z[x]</math>.
* {{math|''α'' ∈ ''K''}} is an algebraic integer if <math>\Z[\alpha]</math> is a finitely generated <math>\Z</math>-module.
* {{math|''α'' ∈ ''K''}} is an algebraic integer if there exists a non-zero finitely generated <math>\Z</math>-[[submodule]] <math>M \subset \Complex</math> such that {{math|''αM'' ⊆ ''M''}}.

Algebraic integers are a special case of [[integral element]]s of a ring extension. In particular, an algebraic integer is an integral element of a finite extension <math>K / \mathbb{Q}</math>.

==Examples==
* The only algebraic integers that are found in the set of rational numbers are the integers. In other words, the intersection of <math>\mathbb{Q}</math> and {{mvar|A}} is exactly <math>\mathbb{Z}</math>. The rational number {{math|{{sfrac|''a''|''b''}}}} is not an algebraic integer unless {{mvar|b}} [[divisor|divides]] {{mvar|a}}. The leading coefficient of the polynomial {{math|''bx'' − ''a''}} is the integer {{mvar|b}}.
* The [[square root]] <math>\sqrt{n}</math> of a nonnegative integer {{mvar|n}} is an algebraic integer, but is [[irrational number|irrational]] unless {{mvar|n}} is a [[square number|perfect square]].
*If {{mvar|d}} is a [[square-free integer]] then the [[field extension|extension]] <math>K = \mathbb{Q}(\sqrt{d}\,)</math> is a [[quadratic field extension|quadratic field]] of rational numbers. The ring of algebraic integers {{math|{{mathcal|O}}<sub>''K''</sub>}} contains <math>\sqrt{d}</math> since this is a root of the monic polynomial {{math|''x''<sup>2</sup> − ''d''}}. Moreover, if {{math|''d'' ≡ 1 [[modular arithmetic|mod]] 4}}, then the element <math display=inline>\frac{1}{2}(1 + \sqrt{d}\,)</math> is also an algebraic integer. It satisfies the polynomial {{math|''x''<sup>2</sup> − ''x'' + {{sfrac|1|4}}(1 − ''d'')}} where the [[constant term]] {{math|{{sfrac|1|4}}(1 − ''d'')}} is an integer. The full ring of integers is generated by <math>\sqrt{d}</math> or <math display=inline>\frac{1}{2}(1 + \sqrt{d}\,)</math> respectively. See [[Quadratic integer]] for more.
*The ring of integers of the field <math>F = \Q[\alpha]</math>, {{math|1=''α'' = {{radic|''m''|3}}}}, has the following [[integral basis]], writing {{math|1=''m'' = ''hk''<sup>2</sup>}} for two [[square-free integer|square-free]] [[coprime]] integers {{mvar|h}} and {{mvar|k}}:<ref>{{cite book| last1=Marcus | first1=Daniel A. | title=Number Fields |edition=3rd | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90279-1 | year=1977 |at=ch. 2, p. 38 and ex. 41}}</ref> <math display="block">\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}</math>
* If {{mvar|ζ<sub>n</sub>}} is a [[primitive root of unity|primitive]] {{mvar|n}}th [[root of unity]], then the ring of integers of the [[cyclotomic field]] <math>\Q(\zeta_n)</math> is precisely <math>\Z[\zeta_n]</math>.
* If {{mvar|α}} is an algebraic integer then {{math|1=''β'' = {{radic|''α''|''n''}}}} is another algebraic integer. A polynomial for {{mvar|β}} is obtained by substituting {{math|''x<sup>n</sup>''}} in the polynomial for {{mvar|α}}.

==Non-example==
* If {{math|''P''(''x'')}} is a [[Primitive polynomial (ring theory)|primitive polynomial]] that has integer coefficients but is not monic, and {{mvar|P}} is [[irreducible polynomial|irreducible]] over <math>\mathbb{Q}</math>, then none of the roots of {{mvar|P}} are algebraic integers (but ''are'' [[algebraic number]]s). Here ''primitive'' is used in the sense that the [[highest common factor]] of the coefficients of {{mvar|P}} is 1, which is weaker than requiring the coefficients to be pairwise relatively prime.

==Finite generation of ring extension==
For any {{math|&alpha;}}, the [[Subring#Ring_extensions|ring extension]] (in the sense that is equivalent to [[field extension]]) of the integers by {{math|&alpha;}}, denoted by <math>\Z[\alpha] \equiv \left\{\sum_{i=0}^n \alpha^i z_i | z_i\in \Z, n\in \Z\right\}</math>, is [[Finitely generated abelian group|finitely generated]] if and only if {{math|&alpha;}} is an algebraic integer.

The proof is analogous to that of the [[Algebraic_number#Degree_of_simple_extensions_of_the_rationals_as_a_criterion_to_algebraicity|corresponding fact]] regarding [[algebraic number]]s, with <math>\Q</math> there replaced by <math>\Z</math> here, and the notion of [[Degree of a field extension|field extension degree]] replaced by finite generation (using the fact that <math>\Z</math> is finitely generated itself); the only required change is that only non-negative powers of {{math|&alpha;}} 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>\Z</math> or <math>\Q</math>, respectively.

==Ring==
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 [[Ring (mathematics)|ring]].

This can be shown analogously to [[Algebraic_number#Field|the corresponding proof]] for [[algebraic number]]s, using the integers <math>\Z</math> instead of the rationals <math>\Q</math>.

One may also construct explicitly the monic polynomial involved, which is generally of higher [[degree of a polynomial|degree]] than those of the original algebraic integers, by taking [[resultant]]s and factoring. For example, if {{math|1=''x''<sup>2</sup> − ''x'' − 1 = 0}}, {{math|1=''y''<sup>3</sup> − ''y'' − 1 = 0}} and {{math|''z'' {{=}} ''xy''}}, then eliminating {{mvar|x}} and {{mvar|y}} from {{math|1=''z'' − ''xy'' = 0}} and the polynomials satisfied by {{mvar|x}} and {{mvar|y}} using the resultant gives {{math|1=''z''<sup>6</sup> − 3''z''<sup>4</sup> − 4''z''<sup>3</sup> + ''z''<sup>2</sup> + ''z'' − 1 = 0}}, which is irreducible, and is the monic equation satisfied by the product. (To see that the {{mvar|xy}} is a root of the {{mvar|x}}-resultant of {{math|''z'' − ''xy''}} and {{math|''x''<sup>2</sup> − ''x'' − 1}}, one might use the fact that the resultant is contained in the [[ideal (ring theory)|ideal]] generated by its two input polynomials.)

===Integral closure===
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 [[integrally closed domain|integrally closed]] in any of its extensions.

Again, the proof is analogous to [[Algebraic_number#Algebraic_closure|the corresponding proof]] for [[algebraic number]]s being [[algebraically closed field|algebraically closed]].

==Additional facts==
* 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 [[quintic]]s are not. This is the [[Abel–Ruffini theorem]]. <!-- what is the meaning of "most" roots of irreducible quintics? By counting, there are as many non-solvable as solvable quintics. Are coefficients of the quintic taken "randomly" from the integers? There ain't no such "random" integer! //--><!--How about this: Consider irreducible quintics of degree n, with integer coefficients with absolute value <= a. Does the proportion of them that are solvable not approach 0 as n and a go to infinity, whether separately or together?-->
* The ring of algebraic integers is a [[Bézout domain]], as a consequence of the [[principal ideal theorem]].
* If the monic polynomial associated with an algebraic integer has constant term 1 or −1, then the [[multiplicative inverse|reciprocal]] of that algebraic integer is also an algebraic integer, and each is a [[unit (ring theory)|unit]], an element of the [[group of units]] of the ring of algebraic integers.
* If {{math|''x''}} is an algebraic number then {{math|''a''<sub>''n''</sub>''x''}} is an algebraic integer, where {{mvar|x}} satisfies a polynomial {{math|''p''(''x'')}} with integer coefficients and where {{math|''a''<sub>''n''</sub>''x''<sup>''n''</sup>}} is the highest-degree term of {{math|''p''(''x'')}}.  The value {{math|1=''y'' = ''a''<sub>''n''</sub>''x''}} is an algebraic integer because it is a root of {{math|1=''q''(''y'') = ''a''{{su|b=''n''|p=''n''&thinsp;−&thinsp;1}}&thinsp;''p''(''y''{{hairsp}}/''a''<sub>''n''</sub>)}}, where {{math|''q''(''y'')}} is a monic polynomial with integer coefficients.
* If {{math|''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 {{math|{{abs|''a''<sub>''n''</sub>}}''x'' / {{abs|''a''<sub>''n''</sub>}}}}, where {{mvar|x}} satisfies a polynomial {{math|''p''(''x'')}} with integer coefficients and where {{math|''a''<sub>''n''</sub>''x''<sup>''n''</sup>}} is the highest-degree term of {{math|''p''(''x'')}}.
* The only rational algebraic integers are the integers. That is, if {{mvar|x}} is an algebraic integer and <math>x\in\Q</math> then <math>x\in\Z</math>. This is a direct result of the [[rational root theorem]] for the case of a monic polynomial.

==See also==
*[[Gaussian integer]]
*[[Eisenstein integer]]
*[[Root of unity]]
*[[Dirichlet's unit theorem]]
*[[Fundamental unit (number theory)|Fundamental units]]

==References==
<references />
{{refbegin}}
* {{cite book|first=William |last=Stein |authorlink = William A. Stein|title=Algebraic Number Theory: A Computational Approach |url=http://wstein.org/books/ant/ant.pdf |archive-url=https://web.archive.org/web/20131102070632/http://wstein.org/books/ant/ant.pdf |archive-date=2013-11-02 |url-status=live}}
{{refend}}
{{Algebraic numbers}}

[[Category:Algebraic numbers]]
[[Category:Integers]]