## Wednesday 15 June 2011 - Dajano Tossici

**ARGOMENTI:** Seminars Ph.D. Program

SEMINARIO DOTTORATO

Wednesday 15 June 2011 h. 14:30, room 2BC30

Dajano Tossici (Univ. Milano-Bicocca)

"On the essential dimension of groups"

-Abstract

At the end of the nineteenth century many authors (Klein, Hermite, Hilbert for instance) studied the problem of reducing the number of parameters of a generic polynomial of fixed degree. This problem was motivated by the problem of finding a formula, in terms of the usual algebraic operations and of radicals, for the roots of polynomial equations. It is very well known that, later, Galois proved that for polynomial equations of degree greater than 5 this formula does not exists.

In 1997 Buhler and Reichstein rewrote and generalized this problem in a more modern context. They introduced the notion of essential dimension of a finite group G, which, very roughly speaking, computes the number of parameters needed to describe all Galois extensions with Galois group G. If we consider the symmetric group S_n then one obtains the number of parameters needed to write a generic polynomial of degree n.

In the talk, after recalling the classical problem described above and the precise definition of essential dimension of a group, we illustrate several examples and open problems. At the very end of the talk, if time is left, we quickly give an overview of results we obtained in collaboration with Angelo Vistoli about essential dimension of a group scheme, which is a generalization of the concept of group in the context of algebraic geometry.

Rif. int. C. Marastoni, T. Vargiolu

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