S-unit group (${\mathcal{O}}_{K,S}^{\times }$)

For a number field $K$ and a finite set $S$ of finite primes, an S-unit is an element of $K^{\times }$ whose discrete valuation is zero at every prime outside $S$. Its group is the unit group of the ring obtained by allowing denominators at $S$. The Dirichlet unit theorem and the valuation homomorphism to ${\mathbb{Z}}^S$ show that this group is a finitely generated abelian group.