PlanetMath: Dirichlet's unit theorem

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Dirichlet's unit theorem

(Theorem)

Let be a number field, and let $\mathcal{O}_K$ be its ring of integers. Then

$\displaystyle \mathcal{O}_K^*\cong \mu(K)\times\mathbb{Z}^{r+s-1}.$

Here $\mathcal{O}_K^*$ is the group of units of $\mathcal{O}_K$ , $\mu(K)$ is the finite cyclic group of the roots of unity in $\mathcal{O}_K^*$ ,

is the number of real embeddings $K\rightarrow \mathbb{R}$ , and

is the number of non-real complex embeddings $K\rightarrow \mathbb{C}$ (which occur in complex conjugate pairs, so

is an integer).

"Dirichlet's unit theorem" is owned by yark. [ full author list (2) | owner history (1) ]

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See Also: regulator

Attachments:: units of quadratic fields (Application) by pahio
units of real cubic fields with exactly one real embedding (Application) by Wkbj79
characterizing CM-fields using Dirichlet's unit theorem (Theorem) by rm50

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Cross-references: integer, complex conjugate, complex embeddings, real embeddings, roots of unity, cyclic group, finite, group of units, ring of integers, number field
There are 7 references to this entry.

This is version 7 of Dirichlet's unit theorem, born on 2003-01-20, modified 2007-12-14.
Object id is 3911, canonical name is DirichletsUnitTheorem.
Accessed 2974 times total.

Classification:

AMS MSC:	11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)
	11R27 (Number theory :: Algebraic number theory: global fields :: Units and factorization)

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