• Hasse-Arf theorem
• local fields
• ramification groups
• ramification jumps

The Hasse-Arf theorem is a classical result of algebraic number theory which state that if $L|K$ is a finite abelian Galois extension of local fields, then the upper jumps of $L|K$ are integers. The aim of this thesis is to give a proof of this theorem, following the book "Local fields"(Serre, 1979), and a proof of its converse, following the article "A converse to the Hasse-Arf theorem"(Elder and Keating,2023) . The converse state that if $G$ is a non abelian group which is isomorphic to the Galois group of some totally ramified extension $E|F$ of local field with residue characteristic $p >2$, then there is a totally ramified extension of local field, with residue characteristic $p$, such that $Gal(L|K) \cong G$ and $L|K$ has at leat one nonintegral upper ramification jump.



In the first chapter, I have gathered various results from algebraic number theory and Galois theory that are necessary to understand the findings in the rest of the thesis.

In the second chapter, a proof of the Hasse-Arf theorem is presented, with a central focus on the study of the norm operator in the context of totally ramified extensions of local fields.

The third chapter covers several key topics for proving the converse of the Hasse-Arf theorem, including Artin-Schreier theory in the setting of local fields and some significant properties of finite p-groups.

Finally, in the fourth and last chapter, we prove the converse of the Hasse-Arf theorem, first in the case where G is a p-group and then in the more general setting. We conclude with some results on the converse in the case where G is the Galois group of an extension of local fields with a residue field of characteristic 2.