##### Department of Mathematics,

University of California San Diego

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### Representation Theory

## Hanspeter Kraft

#### University of Basel, Switzerland

## Equations for field extensions and covariants of the symmetric

##### Abstract:

For a given finite separable field extension $L/K$ we would like to find a generator $x \in L$ whose equation is as simple as possible. For example, one would like to have as many vanishing coefficients as possible. More generally, the transcendence degree of the subfield $k$ of $K$ generated by the coefficients of the equation should be as small as possible. It was shown by Buhler and Reichstein that the minimal transcendence degree one can reach has an interpretation in terms of rational covariants of the symmetric group $S_n$ where $n := [L:K]$. This number is called the essential dimension of $L/K$ or of $S_n$ it can be defined for every finite group. We will describe the relation between covariants and equations for field extensions and will explain the main results in this context.

Host: K. Baur

### September 28, 2004

### 2:00 PM

### AP&M 7321

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