Diophantine Geometry: An Introduction
《Diophantine Geometry: An Introduction》 · 第 273–274 页 · Chapter B，Section 5「The Dirichlet Unit Theorem」，Theorem B.5.1
Theorem C.3.3. (Dirichlet unit theorem) Let $k$ be a number field, let $r_1$ and $r_2$ be respectively the number of its real and complex archimedean places, and let $S$ be a finite set of nonarchimedean places of $k$. Then the group of $S$-units $R_{k,S}^*$ is a finitely generated group of rank $r_1+r_2-1+\#S$.