Integral Hopf–Galois Structures on Degree p² Extensions of p-adic Fields

Abstract

Let L/K be a totally ramified, normal extension of p-adic fields of degree p². We investigate the behavior of the valuation ring $\text{O}$_L in the various Hopf–Galois structures on L/K. Specifically, we determine when $\text{O}$_L is Hopf–Galois with respect to a Hopf order in the corresponding Hopf algebra. When this occurs, $\text{O}$_L is necessarily a free module over this Hopf order. We also determine which Hopf orders can arise in this way. For cyclic extensions L/K of degree p², L. N. Childs has shown, under certain restrictions on the ramification numbers, that if $\text{O}$_L is Hopf–Galois with respect to a Hopf order in one of the Hopf–Galois structures on L/K, then the same is true in all p Hopf–Galois structures on L/K. We show that this no longer holds if the ramification conditions are relaxed, or if elementary abelian extensions of degree of p² are considered. We illustrate our results with a special family of Kummer extensions, and with certain extensions arising from Lubin–Tate formal groups.