Minimal ramification in nilpotent extensions

Nadya Markin and Stephen V. Ullom

Vol. 253 (2011), No. 1, 125–143

DOI: 10.2140/pjm.2011.253.125

Abstract

Let G be a finite nilpotent group and K a number field with torsion relatively prime to the order of G. By a sequence of central group extensions with cyclic kernel we obtain an upper bound for the minimum number of prime ideals of K ramified in a Galois extension of K with Galois group isomorphic to G. This sharpens and extends results of Geyer and Jarden and of Plans. Alternatively, we show how to use Fröhlich’s result on realizing the Schur multiplicator in order to realize a family of groups given by central extensions with minimal ramification.

Keywords

class field theory, inverse Galois theory, nilpotent groups

Mathematical Subject Classification 2000

Primary: 12F12, 11S31, 12F10, 11R32

Milestones

Received: 9 March 2010

Accepted: 21 December 2010

Published: 28 November 2011

Proposed: Jonathan D. Rogawski