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A generalization of Dirichlet's unit theorem

Volume 162 / 2014

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


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 $\mathbb {R}$.


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

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