TY - JOUR

T1 - Subgroups of Hol Q8 as Galois groups

AU - Ledet, Arne

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1996/4/15

Y1 - 1996/4/15

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0001262424&partnerID=8YFLogxK

U2 - 10.1006/jabr.1996.0130

DO - 10.1006/jabr.1996.0130

M3 - Article

AN - SCOPUS:0001262424

VL - 181

SP - 478

EP - 506

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 2

ER -