#### J. Buhler, Z. Reichstein,
* On the Essential Dimension of a Finite Group *,
Compositio Math. 106 (1997), 159-179.

** Abstract: **
Let f(x) = \sum a_i x^i be a monic polynomial of degree n whose
coefficients are algebraically independent variables over a base
field k of characteristic 0. We say that a polynomial g(x) is
generating (for the symmetric group) if it can be obtained
from f(x) by a non-degenerate Tschirnhaus transformation.
We show that the minimal number d_k(n) of algebraically independent
coefficients of such a polynomial is at least [n/2]. This generalizes
a classical theorem of Felix Klein on quintic polynomials and is related
to an algebraic form of Hilbert's 13-th problem.

Our approach to this question (and generalizations) is based
on the idea of the ``essential dimension'' of a finite group G:
the smallest possible dimension of an algebraic G-variety over k
to which one can ``compress'' a faithful linear representation of G.
We show that d_k(n) is just the essential dimension of the symmetric
group S_n. We give results on the essential dimension of
other groups. In the last section we relate the notion of essential
dimension to versal polynomials and discuss their relationship to
the generic polynomials of Kuyk, Saltman and DeMeyer.

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