• Galois module structure
• Hopf Galois

In this thesis we present a generalization of Leopoldt theorem for Galois module structure in the $p$-adic case, due to Lettl:
if $L/\Q_p$ is abelian, then $\Ol$ is free over the associated order $\A_{L/K}$ for each $K⊆L$.
The proof of this theorem, that forms the main body of the second chapter of the thesis, is split into two parts:
first we prove it in the case when $L⊆\Q_p(ζ_{p^n})$, where we also provide an explicit description of the associated order and a generator for $\Ol$ over it,
and then using structural lemmas we extend the result to the general case.

In the third and last chapter of this thesis, after introducing Hopf-Galois theory, we prove a generalization of Noether's theorem due to Childs: if the associated order is a Hopf order, then $\Ol$ is free over it, both in the classical case and in the more general Hopf-Galois case.
Then, using Lubin-Tate formal groups, we produce a class of examples, due to Byott, of $p$-adic field extensions $L/K$
that have both a Galois and a non-clasical Hopf-Galois structure, such that $\Ol$ is free over the non-classical associated order, but not over the classical one.
This further confirms the value of considering non-classical Hopf-Galois extensions.