Algebra & Number Theory

Realizing 2-groups as Galois groups following Shafarevich and Serre

Peter Schmid

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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

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

Received: 26 July 2017
Revised: 21 July 2018
Accepted: 26 August 2018
First available in Project Euclid: 14 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Galois 2-groups Scholz fields tame ramification Shafarevich Serre


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.

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