Galois groups of order 2n that contain a cyclic subgroup of order n

Y.-S. Hwang, David B. Leep and Adrian R. Wadsworth

Vol. 212 (2003), No. 2, 297–319

DOI: 10.2140/pjm.2003.212.297

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 n^th 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).

Milestones

Received: 10 October 2002

Revised: 26 December 2002

Published: 1 December 2003