Editing Algebraic integer (section)

{{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]].