PlanetMath: the ring of integers of a number field is finitely generated over $\mathbb{Z}$

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (210)
Orphanage
Unclass'd (1)
Unproven (473)
Corrections (34)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

the ring of integers of a number field is finitely generated over $\mathbb{Z}$

(Theorem)

Theorem 1 Let be a number field of degree over $\mathbb{Q}$ and let $\mathcal{O}_K$ be the ring of integers of . The ring $\mathcal{O}_K$ is a free abelian group of rank . In other words, there exists a finite integral basis (with elements) for , i.e. there exist algebraic integers $\alpha_1,\ldots,\ \alpha_n$ such that every element of $\mathcal{O}_K$ can be expressed uniquely as a $\mathbb{Z}$ -linear combination of the $\alpha_i$ .

Corollary 1 Every ideal of $\mathcal{O}_K$ is finitely generated.

Proof. [Proof of the corollary] By the theorem, $\mathcal{O}_K$ is a free abelian group of rank

, and therefore it is finitely generated. Notice that an ideal is an additive subgroup. Finally a subgroup of a finitely generated free abelian group is also finitely generated. $\qedsymbol$

This is the first step to prove that $\mathcal{O}_K$ is a Dedekind domain. Notice that the field of fractions of $\mathcal{O}_K$ is the field itself. Therefore, by definition, $\mathcal{O}_K$ is integrally closed in .

"the ring of integers of a number field is finitely generated over $\mathbb{Z}$ " is owned by alozano.

(view preamble)

This object's parent.

Attachments:: proof of the ring of integers of a number field is finitely generated over $\mathbb{Z}$ (Proof) by rm50
integral closures in separable extensions are finitely generated (Theorem) by rm50

Log in to rate this entry.
(view current ratings)

Cross-references: integrally closed, field, field of fractions, Dedekind domain, subgroup, additive, finitely generated, ideal, combination, algebraic integers, integral basis, finite, rank, free abelian group, ring, ring of integers, degree, number field
There is 1 reference to this entry.

This is version 4 of the ring of integers of a number field is finitely generated over $\mathbb{Z}$ , born on 2005-03-17, modified 2005-03-17.
Object id is 6883, canonical name is RingOfIntegersOfANumberFieldIsFinitelyGeneratedOverMathbbZ.
Accessed 1324 times total.

Classification:

AMS MSC:

13B22 (Commutative rings and algebras :: Ring extensions and related topics :: Integral closure of rings and ideals ; integrally closed rings, related rings )

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact