The Normal Closures of Certain Kummer Extensions

Abstract

AbstractLet 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.