New York Journal of Mathematics
Volume 2 (1996) 86-102


Lindsay N. Childs

Hopf Galois Structures on Degree p2 Cyclic Extensions of Local Fields

Published: December 5, 1996
Keywords: Galois module, Hopf Galois extension, associated order, wildly ramified, Hopf order
Subject: 11S15, 11R33, 16W30

Let L be a Galois extension of K, finite field extensions of Qp, p odd, with Galois group cyclic of order p2. There are p distinct K-Hopf algebras Ad, d = 0,...,p-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 Ad, and find criteria on Ev for Ev to be 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 Galois over R for some Ev, and show that if S is Ev-Hopf Galois over R for some Ev, then the associated order Ad of S in Ad is Hopf, and hence S is Ad-free, for all d. Finally we parametrize 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.

Author information

Department of Mathematics and Statistics, University at Albany, Albany, NY 12222