• extension
• Galois
• Hopf
• integral structures
• number theory
• teoria algebrica dei numeri

The thesis is organized as follows.

In Chapter 1 we introduce the basic definitions and results
on Hopf algebras and define the Hopf Galois extensions following \cite{childs2000taming}. In Chapter 2 we present the already mentioned theorem by
Greither and Pareigis and the so called Byott's translation. As an application of these results, we count the number of different Hopf Galois structure that a cyclic Galois extension of degree a power of a prime may have. Main sources for
this chapter are \cite{childs2000taming}, \cite{pareigis1987hopf} and \cite{Byott1996UniquenessOH}. In Chapter 3, after introducing the associated order and the concept of tame $H$-extension, we prove the generalization of Noether's theorem due to Childs. In the last chapter, we consider a totally ramified normal extension $L/K$ of $p$-adic fields with degree $p^2$ and we investigate for which Hopf-Galois structure over $L/K$ we have that $\mathcal{O}_L$ is Hopf-Galois following \cite{Byott2002IntegralHS} by Byott.
We start finding some arithmetic necessary conditions on an extension $L/K$ that makes $\mathcal{O}_L$ Hopf Galois: $t_1=pj-1 \leq t_2=p^2i-1$.
The main result is a total characterisation of the behavior of $\mathcal{O}_L$ in the different Hopf Galois structures: under the assumption that $L/K$ has ramification numbers $t_1=pj-1 \leq t_2=p^2i-1$ there are sufficient and necessary conditions on $i$ and $j$ for having $\mathcal{O}_L$ Hopf–Galois with respect to some Hopf–Galois structure on $L/K$.\\