Subgroups of Hol Q8 as Galois groups

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Abstract

In this paper, we consider nonsplit Galois theoretical embedding problems with cyclic kernel of prime order p, in the case where the ground field has characteristic ≠ p. It is shown that such an embedding problem can always be reduced to another embedding problem, in which the ground field contains the primitive pth roots of unity, and the group extension is central. The reduction is effective, in the sense that a solution to the reduced embedding problem induces a solution to the original embedding problem and that all solutions to the original embedding problem are induced in this way from solutions to the reduced embedding problem. The simplest case of this reduction is then used to give criteria for the realisability of four subgroups of the holomorph Hol Q8, where Q8 is the quaternion group of order 8, including the holomorph itself.

Original languageEnglish
Pages (from-to)478-506
Number of pages29
JournalJournal of Algebra
Volume181
Issue number2
DOIs
StatePublished - Apr 15 1996

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