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Transactions of the American Mathematical Society

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

 
 

 

Densities of primes and realization of local extensions


Author: A. B. Ivanov
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11R34, 11R44, 11R45
DOI: https://doi.org/10.1090/tran/7449
Published electronically: April 25, 2018
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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.


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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
Email: ivanov@ma.tum.de, ivanov@math.uni-bonn.de

DOI: https://doi.org/10.1090/tran/7449
Keywords: Number field, Galois cohomology, restricted ramification, Dirichlet density
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
Article copyright: © Copyright 2018 American Mathematical Society

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