### 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 Q_{8}, where Q_{8} is the quaternion group of order 8, including the holomorph itself.

Original language | English |
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Pages (from-to) | 478-506 |

Number of pages | 29 |

Journal | Journal of Algebra |

Volume | 181 |

Issue number | 2 |

DOIs | |

State | Published - Apr 15 1996 |

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## Cite this

_{8}as Galois groups.

*Journal of Algebra*,

*181*(2), 478-506. https://doi.org/10.1006/jabr.1996.0130