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Extra-Special Groups of Order 32 as Galois Groups

Published online by Cambridge University Press:  20 November 2018

Tara L. Smith
Tara L. Smith*
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, U.S.A.
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Abstract

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In this article we examine conditions for the appearance or nonappearance of the two extra-special 2-groups of order 32 as Galois groups over a field F of characteristic not 2. The groups in question are the central products DD of two dihedral groups of order 8, and DQ of a dihedral group with the quaternion group, obtained by identifying the central elements of order 2 in each factor group. It is shown that the realizability of each of these groups as Galois groups over F implies the realizability of other 2-groups (which are not their quotient groups), and in turn that realizability of certain other 2-groups implies the realizability of DD and DQ. We conclude by providing an explicit construction of field extensions with Galois group DD.

Keywords

12F
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 4 , 01 August 1994 , pp. 886 - 896
Copyright
Copyright © Canadian Mathematical Society 1994

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