the ring of integers of a number field is finitely generated over


Let K be a number field of degree n over Q and let OK be the ring of integersMathworldPlanetmath of K. The ring OK is a free abelian groupMathworldPlanetmath of rank n. In other words, there exists a finite integral basis (with n elements) for K, i.e. there exist algebraic integersMathworldPlanetmath α1,,αn such that every element of OK can be expressed uniquely as a Z-linear combinationMathworldPlanetmath of the αi.


Every ideal of OK is finitely generatedMathworldPlanetmathPlanetmath.

Proof of the corollary.

By the theorem, 𝒪K is a free abelian group of rank n, and therefore it is finitely generated. Notice that an ideal is an additive subgroupMathworldPlanetmathPlanetmath. Finally a subgroup of a finitely generated free abelian group is also finitely generated. ∎

This is the first step to prove that 𝒪K is a Dedekind domainMathworldPlanetmath. Notice that the field of fractionsMathworldPlanetmath of 𝒪K is the field K itself. Therefore, by definition, 𝒪K is integrally closedMathworldPlanetmath in K.

Title the ring of integers of a number field is finitely generated over
Canonical name TheRingOfIntegersOfANumberFieldIsFinitelyGeneratedOvermathbbZ
Date of creation 2013-03-22 15:08:22
Last modified on 2013-03-22 15:08:22
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 7
Author alozano (2414)
Entry type Theorem
Classification msc 13B22