Realizing 2-groups as Galois groups following Shafarevich and Serre

Schmid, Peter

doi:10.2140/ant.2018.12.2387

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Realizing 2-groups as Galois groups following Shafarevich and Serre

Peter Schmid

Vol. 12 (2018), No. 10, 2387–2401

DOI: 10.2140/ant.2018.12.2387

Abstract

Let $G$ be a finite $p$ -group for some prime $p$ , say of order $p^{n}$ . For odd $p$ the inverse problem of Galois theory for $G$ has been solved through the (classical) work of Scholz and Reichardt, and Serre has shown that their method leads to fields of realization where at most $n$ rational primes are (tamely) ramified. The approach by Shafarevich, for arbitrary $p$ , has turned out to be quite delicate in the case $p = 2$ . In this paper we treat this exceptional case in the spirit of Serre’s result, bounding the number of ramified primes at least by an integral polynomial in the rank of $G$ , the polynomial depending on the $2$ -class of $G$ .

Keywords

Galois 2-groups, Scholz fields, tame ramification, Shafarevich, Serre

Mathematical Subject Classification 2010

Primary: 11R32

Secondary: 20D15

Milestones

Received: 26 July 2017

Revised: 21 July 2018

Accepted: 26 August 2018

Published: 1 February 2019

Authors

Peter Schmid
Mathematisches Institut Universität Tübingen Tübingen Germany

Vol. 12, No. 10, 2018