Galois scaffolds and Galois module structure in extensions of\n characteristic $p$ local fields of degree $p^2$

A Galois scaffold, in a Galois extension of local fields with perfect residue\nfields, is an adaptation of the normal basis to the valuation of the extension\nfield, and thus can be applied to answer questions of Galois module structure.\nHere we give a sufficient condition for a Galois scaffold to exist in fully\nramified Galois extensions of degree $p^2$ of characteristic $p$ local fields.\nThis condition becomes necessary when we restrict to $p=3$. For extensions\n$L/K$ of degree $p^2$ that satisfy this condition, we determine the Galois\nmodule structure of the ring of integers by finding necessary and sufficient\nconditions for the ring of integers of $L$ to be free over its associated order\nin $K[Gal(L/K)]$.\n