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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Densities of primes and realization of local extensions
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by A. B. Ivanov PDF
Trans. Amer. Math. Soc. 371 (2019), 83-103 Request permission

Abstract:

In this paper we introduce new densities on the set of primes of a number field. If $K/K_0$ is a Galois extension of number fields, we associate to any element $x \in \mathrm {G}_{K/K_0}$ a density $\delta _{K/K_0,x}$ on the primes of $K$. In particular, the density associated to $x = 1$ is the usual Dirichlet density on $K$. We also give two applications of these densities (for $x \neq 1$): the first is a realization result à la the Grunwald-Wang theorem such that essentially, ramification is only allowed in a set of arbitrarily small (positive) Dirichlet density. The second concerns the so-called saturated sets of primes, introduced by Wingberg.
References
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Additional Information
  • A. B. Ivanov
  • Affiliation: Institut de Mathématiques de Jussieu, 4, place Jussieu, 75252 Paris cedex 05, France
  • Address at time of publication: Endenicher Allee 60, 53115 Bonn, Germany
  • MR Author ID: 1014138
  • Email: ivanov@ma.tum.de, ivanov@math.uni-bonn.de
  • Received by editor(s): October 20, 2016
  • Received by editor(s) in revised form: January 12, 2017
  • Published electronically: April 25, 2018
  • Additional Notes: The author was supported by the Technical University Munich and by the HIM center in Bonn
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 371 (2019), 83-103
  • MSC (2010): Primary 11R34, 11R44, 11R45
  • DOI: https://doi.org/10.1090/tran/7449
  • MathSciNet review: 3885138