### Abstract

Let n be any integer with n > 1, and let F ⊆ L be fields such that [L:F] = 2, L is Galois over F, and L contains a primitive nth root of unity ζ. For a cyclic Galois extension M = L(α^{1/n}) of L of degree n such that M is Galois over F, we determine, in terms of the action of Gal(L/F) on α and ζ, what group occurs as Gal(M/F). The general case reduces to that where n = p^{e}, with p prime. For n = p^{e}, we give an explicit parametrization of those α that lead to each possible group Gal(M/F).

Original language | English |
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Pages (from-to) | 297-319 |

Number of pages | 23 |

Journal | Pacific Journal of Mathematics |

Volume | 212 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2003 Dec |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Hwang, Y. S., Leep, D. B., & Wadsworth, A. R. (2003). Galois groups of order 2n that contain a cyclic subgroup of order n.

*Pacific Journal of Mathematics*,*212*(2), 297-319. https://doi.org/10.2140/pjm.2003.212.297