Classification of the Hopf Galois structures on prime power radical extensions (Q1270046)

Let \(k\) be a field of characteristic zero and \(K= k(w)\) a field extension generated by an element \(w\) with \(w^{p^n}= a\in k\), \(p\) a prime. Furthermore assume that \([K:k]= p^n\). Such an extension is finite and separable, but not (in general) Galois. It is, however, Hopf Galois with a Hopf algebra \(H\) which is a \(\widetilde{K}\)-Hopf algebra form of a group ring \(kN\), where \(\widetilde{K}\) is the normal closure of \(K/k\). The paper classifies the different structures of Hopf Galois extensions, in particular the possible Hopf algebras \(H\), depending on the largest integer \(r\) such that \(K\) contains a primitive \(p^r\)-th root of unity. If \(r< n\), the author shows that there are \(p^r\) Hopf Galois structures on \(K/k\).