Scaffolds and integral Hopf Galois module structure on purely inseparable extensions

Let p be prime. Let L/K be a finite, totally ramified, purely inseparable extension of local fields, [L : K] = p, n ≥ 2. It is known that L/K is Hopf Galois for numerous Hopf algebras H, each of which can act on the extension in numerous ways. For a certain collection of such H we construct “Hopf Galois scaffolds” which allow us to obtain a Hopf analogue to the Normal Basis Theorem for L/K. The existence of a scaffold structure depends on the chosen action of H on L. We apply the theory of scaffolds to describe when the fractional ideals of L are free over their associated orders in H.