Equidistribution of realizable Steinitz classes for cyclic Kummer extensions

Let be prime, and K be a number field containing the -th roots of unity. We use classical algebraic number theory and some analytic techniques to prove that the Steinitz classes of Z/ Z extensions of K ordered by relative discriminant are equidistributed among realizable classes in the ideal class group of K. For = 2, this was proved by Kable and Wright using the deep theory of prehomogeneous vector spaces. Foster proved that Steinitz classes are uniformly distributed between realizable classes for tamely ramified elementary-m extensions using the theory of Galois modules; our approach eliminates this tameness hypothesis.