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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).