Taming Wild Extensions with Hopf Algebras

Abstract

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