A generalization of Dirichlet's unit theorem

TL;DR

This paper extends Dirichlet's $S$-unit theorem to an infinite rank group of algebraic numbers with valuations only over a specified set of places, revealing a vector space structure governed by the product formula.

Contribution

It generalizes the classical theorem to an infinite rank setting, showing the algebraic $S$-units form a $Q$-vector space with a hyperplane structure under the Weil height.

Findings

The group of algebraic $S$-units modulo torsion is a $Q$-vector space.

This vector space spans a hyperplane determined by the product formula.

Linearly independent elements over $Q$ remain independent over $R$.

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 $\bQ$-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 $\mathbb{R}$.