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 L1/K, L2/K such that L1∩M2=L2∩M1=K, where Mi is a field such that LiMi=L˜i, the normal closure of Li/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 OL over its associated order in an almost classically Galois structure on L/K.