Equidistribution of realizable Steinitz classes for cyclic Kummer extensions

Let $\mathrm{\ell }$ be prime, and $K$ be a number field containing the $\mathrm{\ell }$-th roots of unity. We use classical algebraic number theory and some analytic techniques to prove that the Steinitz classes of $\mathbb{Z}\mathrm{/}\mathrm{\ell }\mathbb{Z}$ extensions of $K$ ordered by relative discriminant are equidistributed among realizable classes in the ideal class group of $K$. For $\mathrm{\ell }=2$, 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.