A generalization of Kummer theory to Hopf-Galois extensions

Abstract

We introduce a condition for Hopf-Galois extensions that generalizes the notion of Kummer Galois extension. Namely, an $H$-Galois extension $L/K$ is $H$-Kummer if $L$ can be generated by adjoining to $K$ a finite set $S$ of eigenvectors for the action of the Hopf algebra $H$ on $L$. This extends the classical Kummer condition for the classical Galois structure. With this new perspective, we shall characterize a class of $H$-Kummer extensions $L/K$ as radical extensions that are linearly disjoint with the $n$-th cyclotomic extension of $K$. This result generalizes the description of Kummer Galois extensions as radical extensions of a field containing the $n$-th roots of the unity. The main tool is the construction of a product Hopf-Galois structure on the compositum of almost classically Galois extensions $L_1/K$, $L_2/K$ such that $L_1\cap M_2=L_2\cap M_1=K$, where $M_i$ is a field such that $L_iM_i=\widetilde{L}_i$, the normal closure of $L_i/K$. When $L/K$ is an extension of number or $p$-adic fields, we shall derive criteria on the freeness of the ring of integers $\mathcal{O}_L$ over its associated order in an almost classically Galois structure on $L/K$.