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# X. Caruso

## The field of p-adic numbers and its finite extensions

### Краткое описание курса

The absolute Galois group of Q, denoted Gal(\bar{Q}/Q) in the sequel,
is a fundamental object in arithmetics since it carries a lot of
informations about number fields. Unfortunately, it is nowadays not
so well understood.

A classical way to study this group is to use p-adic numbers. Indeed,
for each prime number p, the group Gal(\bar{Q_p}/Q_p) naturally embeds
in Gal(\bar{Q}/Q) on the one hand, and can be more easily described on
the other hand.

In this course, we will present two different aspects of the study of
Gal(\bar{Q_p}/Q_p):

- firstly, we will explain the classical dévissage of
Gal(\bar{Q_p}/Q_p) based on ramification
theory; as a corollary, we will deduce that any finite extension
of Q_p is resoluble;
- secondly, based on previous results, we will focus on (continuous)
representations of $Gal(\bar{Q_p}/Q_p) (in various coefficient rings).