## 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