Exercise 19, § 10

Let E be a Galois extension of a field K with Galois group G. For each $x \in E$ denote by Gx the set of conjugates of x in E; this is a finite G-set. Let x, y be two elements of E. For the G-sets Gx and Gy to be isomorphic (that is, for a bijection Gx → Gy to exist, compatible with the operation of G on the two sets), it is necessary and sufficient that the extensions K(x) and K(y) should be conjugate.