Editing Algebraic integer (section)

==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.