Hopf Galois Structures on Degree p2 Cyclic Extensions of Local Fields
نویسنده
چکیده
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 ≡ −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.
منابع مشابه
Hopf Galois structures on Kummer extensions of prime power degree
Let K be a field of characteristic not p (an odd prime), containing a primitive p-th root of unity ζ, and let L = K[z] with x n − a the minimal polynomial of z over K: thus L|K is a Kummer extension, with cyclic Galois group G = 〈σ〉 acting on L via σ(z) = ζz. T. Kohl, 1998, showed that L|K has pn−1 Hopf Galois structures. In this paper we describe these Hopf Galois structures.
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