Algebraic number theory.
This lecture (4 hrs) will be taught in presence in the room A 027 (Tuesday, Thursday 10:00-12:00).
We will give an introduction to classical algebraic number theory, which means the study of the ring of algebraic integers OK in a number field K (a finite extension of the field of rational number Q ). We will start with some general facts on Dedekind rings, and will then specialize to the rings of the form OK . We will give the proof of classical facts, like the Dirichlet theorem on the structure of the group of units of OK , as well as the finiteness of the class group also due to Dirichlet. We will then introduce and study the ramification in a finite extension K < L , and local phenomena. We will prove the fundamental result due to Hermite-Minkowski that any non trivial number field admits nontrivial ramification with respect to Q. We will finish by studying the behaviour of rings of integers in Ga- lois extensions, and aim to prove the famous Theorem of Kronecker-Weber classifying abelian extensions of Q.
References: P. Samuel, Theorie algebrique des nombres (french, also available in english version under "algebraic number theory").
J.-P. Serre, Corps locaux (french, also available in english version under "local fields").
S. Lang, algebraic number theory.
Exercises classes (Tom Bachmann).
Trees, groups and amalgams.
This lecture (2 hrs) will be taught in presence in the room B 040 (Tuesday 16:00-18:00).
The aim of this lecture is to give an introduction to combinatorial groups theory in the following sense: given a group G acting on a combinatorial graph, what can we say on the structure of G if we know for instance the structure of each of the isotropy subgroups of the action of G at each vertex of the graph ? We will emphasize the case of actions of groups on ``trees" and will give several applications. For instance any subgroup of a free group is a free group. Other arithmetical applications will involve the group SL2 acting on ``its" tree that will be introduced in the second part of the lecture.
Reference: Arbres, Amalgames, SL2, by J.-P. Serre, Astérisque 1977, also available in english:
Trees, by J.-P. Serre, Springer.
Previous semesters .