 

Lindsay N. Childs
Hopf Galois Structures on Degree p^{2} 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 


Abstract
Let L be a Galois extension of K, finite field extensions of
Q_{p}, p odd, with Galois group cyclic of order p^{2}. There are p
distinct KHopf algebras A_{d}, d = 0,...,p1, 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 RHopf orders E_{v} in A_{d}, and find criteria on E_{v} for E_{v} 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 E_{v}Hopf Galois
over R for some E_{v}, and show that if S is E_{v}Hopf Galois over
R for some E_{v}, 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 congruent to
1 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.


Author information
Department of Mathematics and Statistics, University at Albany, Albany, NY 12222
http://math.albany.edu:8000/~lc802/

