Dirichlet's unit theoremDirichletsUnitTheorem2003-01-20 19:53:232007-12-14 09:49:05Theorem\usepackage{amssymb}
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Let $K$ be a number field, and let $\mathcal{O}_K$ be its ring of integers.
Then
\[
\mathcal{O}_K^*\cong \mu(K)\times\mathbb{Z}^{r+s-1}.
\]
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 $\mathcal{O}_K^*$,
$r$ is the number of real embeddings $K\rightarrow \mathbb{R}$,
and $2s$ is the number of non-real complex embeddings $K\rightarrow \mathbb{C}$ (which occur in complex conjugate pairs, so $s$ is an integer).