1Department of Mathematics University of Rochester Rochester, NY 14627, U.S.A. 2Department of Mathematics University of Texas at Austin Austin, TX 78712, U.S.A.
Acta Arithmetica, Tome 162 (2014) no. 4, pp. 355-368
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 $\mathbb {R}$.
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
Affiliations des auteurs :
Paul Fili
1
;
Zachary Miner
2
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.
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author = {Paul Fili and Zachary Miner},
title = {A generalization of {Dirichlet's} unit theorem},
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doi = {10.4064/aa162-4-3},
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Paul Fili; Zachary Miner. A generalization of Dirichlet's unit theorem. Acta Arithmetica, Tome 162 (2014) no. 4, pp. 355-368. doi: 10.4064/aa162-4-3
