Algebra & Number Theory

Realizing 2-groups as Galois groups following Shafarevich and Serre

Peter Schmid

Abstract

Let $G$ be a finite $p$-group for some prime $p$, say of order $pn$. 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 $G$.

Article information

Source
Algebra Number Theory, Volume 12, Number 10 (2018), 2387-2401.

Dates
Revised: 21 July 2018
Accepted: 26 August 2018
First available in Project Euclid: 14 February 2019

https://projecteuclid.org/euclid.ant/1550113226

Digital Object Identifier
doi:10.2140/ant.2018.12.2387

Mathematical Reviews number (MathSciNet)
MR3911134

Zentralblatt MATH identifier
07026821

Subjects
Primary: 11R32: Galois theory
Secondary: 20D15: Nilpotent groups, $p$-groups

Citation

Schmid, Peter. Realizing 2-groups as Galois groups following Shafarevich and Serre. Algebra Number Theory 12 (2018), no. 10, 2387--2401. doi:10.2140/ant.2018.12.2387. https://projecteuclid.org/euclid.ant/1550113226

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