Hopf Galois Structures on Degree p² Cyclic Extensions of Local Fields

Let L be a Galois extension of K, finite field extensions of Qp , p odd, with Galois group cyclic of order p 2 . There are p distinct K-Hopf algebras A d , d = 0; : : : ; p \Gamma 1, which act on L and make L into a Hopf Galois extension of K. We describe these actions. Let R be the valuation ring of K. We describe a collection of R-Hopf orders Ev in A d , and find criteria on Ev for Ev to be the associated order in A d of the valuation ring S of some L. We find criteria on an extension L=K for S to be Ev-Hopf Galois over R for some Ev , and show that if S is Ev-Hopf Galois over R for some Ev , then the associated order A d of S in A d is Hopf, and hence S is A d -free, for all d. Finally we parametrize the extensions L=K whose ramification numbers are j \Gamma1 (mod p 2 ) and determine the density of the parameters of those L=K for which the associated order of S in KG is Hopf