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On 2-Groups as Galois Groups

Published online by Cambridge University Press:  20 November 2018

Arne Ledet
Arne Ledet*
Affiliation:
Matematisk Institut Universitetsparken 5 DK-2100 Copenhagen Ø Denmark e-mail: ledet@math.ku.dk
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Abstract

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Let L/K be a finite Galois extension in characteristic ≠ 2, and consider a non-split Galois theoretical embedding problem over L/K with cyclic kernel of order 2. In this paper, we prove that if the Galois group of L/K is the direct product of two subgroups, the obstruction to solving the embedding problem can be expressed as the product of the obstructions to related embedding problems over the corresponding subextensions of L/K and certain quaternion algebra factors in the Brauer group of K. In connection with this, the obstructions to realising non-abelian groups of order 8 and 16 as Galois groups over fields of characteristic ≠ 2 are calculated, and these obstructions are used to consider automatic realisations between groups of order 4, 8 and 16.

Keywords

12F12
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 6 , 01 December 1995 , pp. 1253 - 1273
Copyright
Copyright © Canadian Mathematical Society 1995

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