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.
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.
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.