#### Vol. 10, No. 1, 2016

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors' Interests Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
Stable sets of primes in number fields

### Alexander Ivanov

Vol. 10 (2016), No. 1, 1–36
##### Abstract

We define a new class of sets —stable sets —of primes in number fields. For example, Chebotarev sets ${P}_{M∕K}\left(\sigma \right)$, with $M∕K$ Galois and $\sigma \in G\left(M∕K\right)$, are very often stable. These sets have positive (but arbitrarily small) Dirichlet density and they generalize sets with density one in the sense that arithmetic theorems such as certain Hasse principles, the Grunwald–Wang theorem, and Riemann’s existence theorem hold for them. Geometrically, this allows us to give examples of infinite sets $S$ with arbitrarily small positive density such that $Spec{\mathsc{O}}_{\phantom{\rule{0.3em}{0ex}}K,S}$ is a $K\left(\pi ,1\right)$ (simultaneously for all $p$).

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/ant

We have not been able to recognize your IP address 3.215.182.36 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

##### Keywords
number field, Galois cohomology, restricted ramification, Dirichlet density
Primary: 11R34
Secondary: 11R45
##### Milestones
Received: 23 June 2014
Revised: 7 September 2015
Accepted: 23 October 2015
Published: 14 February 2016
##### Authors
 Alexander Ivanov Zentrum Mathematik Technischen Universität Boltzmannstraße 3 D-85747 Garching bei München Germany