Ingénieur 2A

Galois theory emerged in XIX century to study the existence of formulas for solutions of polynomial equation (in terms of the coefficients of the equation). The theory is both powerful and elegant and was the origin of a very large part of modern algebra. Nowadays it is also a very active research field.

The aim of this course is first to introduce basics and tools of general algebra (groups, rings, algebras, quotients, field extensions...) which will allow in the second part of the course to develop Galois theory, as well as some of its most remarkable applications.

Beyond the the interest on the subject for itself, the course aims at being a good introduction to algebra and its applications, in Mathematics and in other fields (for instance Computer science with finite fields, Physics and Chemistry with group theory).

*Prerequisites

Standard linear algebra from the first two years at University.

* Knowledge expected at the end of the course : 

Theoretical knowledge :

- Knowledge of fundamental structures in general algebra.

- Knowledge of fundamental concepts in Galois theory (Galois extensions, Galois group)

- Most important examples (finite fields, cyclotomic extensions solvable extensions).

- Main historical applications (solvable polynomial equations, constructability of regular polygons).

Practical knowledge :

- Handling of fundamental algebraic structures, computation of degrees of extensions.

- Characterization of Galois extensions.

- Computation of Galois groups, method of reduction modulo p.

- Applications of the theory, in particular to number theory and fields theory

* Evaluation : exam at the end of the course.

Language  : French