**Title of the course:** Linear representations of finite groups and cyclotomic fields

**Instructor:** Dr. José Ibrahim Villanueva Gutiérrez

**Institution:** Université de Bordeaux

**Dates:** 31 July – 13 August 2017

**Prerequisites:** Linear algebra, algebra, complex analysis, topology.

**Level:** Graduate, advanced undergraduate, beginning undergraduate

**Abstract:** This course is intended to give an insight of the basis of Iwasawa theory to the students.

In order to do this, we start with a very accesible one-week course on linear representations and finish with p-adic characters.

Then we do a one-week course on cyclotomic fields, we start with the basic notions of number fields and their invariants, such as

the ideal class group. We finish by proving the classic Iwasawa's Theorem, that describes the growth of the order of the p-part

of ideal class groups in some kind of infinite extensions closely related to cyclotomic fields.

The first course should be enough to have a good understanding of the second course.

1st Week

1. Permutations and groups. Basic facts about groups, subgroups, group morphisms and the Cayley theorem.

2. Linear algebra. Definition of a module, submodules and module morphisms. The linear general group.

3. Group Representations. Definition and examples of representations. Invariant subspaces.

4. Isomorphisms and direct sum of representations. Maschke theorem.

5. Character Theory. Ortogonality relations of characters.

6. p-adic characters. The p-adic numbers and their structure.

2nd Week

1. Number fields. The main invariants of a number field and their properties.

2. Cyclotomic fields. Basic results about cyclotomic fields.

3. Dirichlet Characters and ramification. Dirichlet characters and ramification.

4. The Iwasawa algebra. Towers of number fields.

5. Noetherian modules over the Iwasawa algebra. Free and torsion Iwasawa modules. The Structure theorem.

6. Iwasawa Theorem.

**Textbook or/and course webpage :**

1st week: Chapter 1 of Linear representations of finite groups, Jean-Pierre Serre. Also see the classic Representation theory of

Harris and Fulton.

2nd week: Introduction to cyclotomic fields of L-C. Washington, Chapters 1-3 and 13. Also Cyclotomic Fields of S. Lang or the notes on

Iwasawa theory (in french) on my web page.

**Language: **EN