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The Normal Closures of Certain Kummer Extensions

Published online by Cambridge University Press:  20 November 2018

William C. Waterhouse
William C. Waterhouse*
Affiliation:
Department of Mathematics The Pennsylvania State University University Park, Pennsylvania 16802 U.S.A.
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Abstract

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Let F be a field containing a primitive p-th root of unity, let K / F be a cyclic extension with group 〈σ〉 of order pn, and choose a in K. This paper shows how the Galois group of the normal closure of K(a1/p) over F can be determined by computations within K. The key is to define a sequence by applying the operation x ↦ σ(x)/x repeatedly to a. The first appearance of a p-th power determines the degree of the extension and restricts the Galois group to one or two possibilities. A certain expression involving that p-th root and the terms in the sequence up to that point is a p-th root of unity, and the group is finally determined by testing whether that root is 1. When (σ(a)/a G Kp, the results reduce to a theorem of A. A. Albert on cyclic extensions.

Keywords

12F10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 1 , 01 March 1994 , pp. 133 - 139
Copyright
Copyright © Canadian Mathematical Society 1994

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