Editing Algebraic integer (section)

==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>.