# Matematica | Prof. Daniel Balescu | Course on "Galois Theory: from groups and forms to descent theory and extensions" | July 2023

To PhD Students in

Mathematics

UNIFE

and,c.c.

To PhD Students in

Physics

To the Coordinators

PhD Courses in

Mathematics and Physics

To Prof. Claudia Menini

"Galois Theory" Course Organization

UNIFE

Dear All,

we are pleased to announce that

**Prof. Daniel Balacu****University of Bucarest (Romania)**

will be visiting INDAM *(Istituto Nazionale Alta Matematica) *and he will give a

**Blended Course on**

** ****"Galois Theory: from groups and forms to descent theory and extensions"**

** ****The course consists in 8 hours and it is scheduled as follows:**

Tue. 4 July - h. 16:00-18:00 - Room F2 Wed. 6 July - h. 16:00-18:00 - Room F2

Tue. 11 July - h. 16:00-18:00 - Room F3Wed. 13 July - h. 16:00-18:00 - Room F3

Room F2 and F3 are at the Chiostro di Santa Maria delle Grazie - Via Fossato di Mortara, 17/19

**Description of the Course**

The classical Galois theory dates back to 1830, but it took more than 100 years for it to be reformulated (by Artin) in the language of module theory.

Artin's criterion that decides when a field extension K/k is Galois allows to extend the classical Galois theory to Hopf algebras.

This was initiated by Chase and Sweedler in 1969 in the commutative case, the general case being considered by Kreimer and Takeuchi in 1981.

Today Hopf-Galois extensions appear in various branches of mathematics and physics, being also known as dual algebraic versions of non-commutative fiber spaces (the notion of quantum fiber space can be introduced as a module associated with a Hopf-Galois extension).

The purpose of this course is to make the transition from the classical Galois theory to the Hopf-Galois theory and to present some directions of study for the latter. Briefly, the content of the course is as follows:

- Classical Galois theory;

- Extensions of fields, extensions of separable Galois fields without groups, strongly graded rings, cross products, affine group schemes;

- Relevant examples of Hopf algebras;

- Hopf-Galois extensions and examples;

- Descent theory;

- Hopf-Galois theories in various categories.

In order to receive the link to the remote lessons, or for further information about the course, please contact Prof. Claudia Menini (Dept. of Mathematics and Informatics) claudia.menini@unife.it

*All participants have to ask Prof. Menini for proof of attendance, and then they can ask their Tutors for the assignment of credits.*