Bianca Gouthier (IMB)
Introduction to essential dimension
In my seminar I will do an introduction to the concept of essential dimension: roughly speaking, the essential dimension is a measure of how many independent parameters we need to describe some algebraic object. The concept of essential dimension was introduced by Buhler and Reichstein in 1995 and it is linked to an algebraic version of Hilbert’s 13th problem. For a finite group G, the essential dimension measures how much one can compress a faithful representation of G. When G is the symmetric group Sn, the essential dimension tells us how many independent parameters we need to write a generic polynomial of degree n on a field k of characteristic zero; equivalently, the essential dimension of Sn computes the number of parameters needed to write a generating
polynomial for separable field extensions of degree n. This is still an open problem for n ≥8. Surprisingly, the analogue problem for inseparable field extensions has been solved explicitely.