S-unit group
= S-unit group
{c}
{title2=$\mathcal O_{K,S}^{\times}$}
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>.