Equidistribution of realizable Steinitz classes for cyclic Kummer extensions

TL;DR

This paper proves that Steinitz classes of cyclic Kummer extensions over a number field are evenly distributed among realizable classes, using classical and analytic number theory techniques, extending previous results to include wild ramification.

Contribution

It introduces a new approach that removes the tameness restriction in the distribution of Steinitz classes for cyclic Kummer extensions.

Findings

Steinitz classes are equidistributed among realizable classes in the ideal class group.

The approach applies to both tame and wild ramification cases.

Extends previous results by Foster and others to more general extensions.

Abstract

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