Dirichlet’s unit theorem

occur in

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

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

Here ${𝒪}_K^{*}$ is the group of units of ${𝒪}_K$, $\mu (K)$ is the finite cyclic group of the roots of unity in ${𝒪}_K^{*}$, $r$ is the number of real embeddings $K\to ℝ$, and $2s$ is the number of non-real complex embeddings $K\to ℂ$ (which occur in complex conjugate pairs, so $s$ is an integer).

Title	Dirichlet’s unit theorem
Canonical name	DirichletsUnitTheorem
Date of creation	2013-03-22 13:22:42
Last modified on	2013-03-22 13:22:42
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	10
Author	yark (2760)
Entry type	Theorem
Classification	msc 11R04
Classification	msc 11R27
Related topic	Regulator