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# Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

## A generalization of Dirichlet's unit theorem

### Tom 162 / 2014

Acta Arithmetica 162 (2014), 355-368 MSC: 11R04, 11R27, 46E30. DOI: 10.4064/aa162-4-3

#### Streszczenie

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 $\mathbb {R}$.

#### Autorzy

• Paul FiliDepartment of Mathematics
University of Rochester
Rochester, NY 14627, U.S.A.
e-mail
• Zachary MinerDepartment of Mathematics
University of Texas at Austin
Austin, TX 78712, U.S.A.
e-mail

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