Hopf Galois Structures on Degree p2 Cyclic Extensions of Local Fields

摘要

Let L be a Galois extension of K, finite field extensions of Qp, p odd, with Galois group cyclic of order p2. There are pdistinct K-Hopf algebras Ad, d = 0,...,p-1, which act on Land make L into a Hopf Galois extension of K. We describe theseactions. Let R be the valuation ring of K. We describe a collectionof R-Hopf orders Ev in Ad, and find criteria on Ev for Ev tobe the associated order in Ad of the valuation ring S of some L.We find criteria on an extension L/K for S to be Ev-Hopf Galoisover R for some Ev, and show that if S is Ev-Hopf Galois overR for some Ev, then the associated order Ad of S in Adis Hopf, and hence S is Ad-free, for all d. Finally weparametrize the extensions L/K whose ramification numbers are congruent to-1 mod p2 and determine the density of the parameters of those L/K for which the associated order of S in KG is Hopf.