Taming wild extensions with Hopf algebras

Abstract

Let K ⊂ L K \subset L be a Galois extension of number fields with abelian Galois group G G and rings of integers R ⊂ S R \subset S , and let A \mathcal {A} be the order of S S in K G KG . If A \mathcal {A} is a Hopf R R -algebra with operations induced from K G KG , then S S is locally isomorphic to A \mathcal {A} as A \mathcal {A} -module. Criteria are found for A \mathcal {A} to be a Hopf algebra when K = Q K = {\mathbf {Q}} or when L / K L/K is a Kummer extension of prime degree. In the latter case we also obtain a complete classification of orders over R R in L L which are tame or Galois H H -extensions, H H a Hopf order in K G KG , using a generalization of the discriminant.