A generalization of Dirichlet's unit theorem

Paul Fili¹ ; Zachary Miner²

¹ Department of Mathematics University of Rochester Rochester, NY 14627, U.S.A.
² Department of Mathematics University of Texas at Austin Austin, TX 78712, U.S.A.

Acta Arithmetica, Tome 162 (2014) no. 4, pp. 355-368

Résumé

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 $\mathbb {R}$.

Détail

DOI : 10.4064/aa162-4-3

Keywords: generalize dirichlets s unit theorem usual group s units number field infinite rank group algebraic numbers having nontrivial valuations only places lying specifically demonstrate group algebraic s units modulo torsion mathbb vector space which normed weil height spans hyperplane determined product formula elements vector space which linearly independent mathbb retain their linear independence mathbb