ETD

• Galois theory
• isomorphism classes of extensions
• p-adic fields
• ramification theory

The goal of this thesis is to classify a certain kind of extensions of degree a power of p of a p-adic field K. In particular we consider the problem of counting the number of K-isomorphism classes of totally ramified extensions of K of degree p^k, where k is any natural number, having no intermediate fields, and of classifying them according to the Galois group of their normal closure. The principal tool is a result which states that there is a one-to-one correspondence between the isomorphism classes of extensions of degree p^k of K having no intermediate extensions and the irreducible H-submodules of dimension k of F*/F*^p, where F is the composite of certain fixed normal extensions of K and H is its Galois group over K. We used this result to obtain explicit formulae when k is a prime number l and for k=4. Moreover we determine the ramification groups and the discriminant of the composite of all extensions of degree p^l of K having no intermediate fields.