A generalization of Dirichlet's unit theorem
Volume 162 / 2014
Acta Arithmetica 162 (2014), 355-368
MSC: 11R04, 11R27, 46E30.
DOI: 10.4064/aa162-4-3
Abstract
We generalize Dirichlet's $S$-unit theorem from the usual group of $S$-units of a number field $K$ to the infinite rank group of all algebraic numbers having nontrivial valuations only on places lying over $S$. Specifically, we demonstrate that the group of algebraic $S$-units modulo torsion is a $\mathbb {Q}$-vector space which, when normed by the Weil height, spans a hyperplane determined by the product formula, and that the elements of this vector space which are linearly independent over $\mathbb {Q}$ retain their linear independence over $\mathbb {R}$.
