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 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 ℚ retain their linear independence over ℝ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa162-4-3,
author = {Paul Fili and Zachary Miner},
title = {A generalization of Dirichlet's unit theorem},
journal = {Acta Arithmetica},
volume = {166},
year = {2014},
pages = {355-368},
zbl = {1285.11126},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa162-4-3}
}
Paul Fili; Zachary Miner. A generalization of Dirichlet's unit theorem. Acta Arithmetica, Tome 166 (2014) pp. 355-368. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa162-4-3/