Compositio Mathematica

, Volume 106, Issue 2, pp 159–179 | Cite as

On the essential dimension of a finite group



Let f(x) = Σaixi be a monic polynomial of degree n whosecoefficients are algebraically independent variables over a base field k of characteristic 0. We say that a polynomial g(x) isgenerating (for the symmetric group) if it can be obtained from f(x) by a nondegenerate Tschirnhaus transformation. We show that the minimal number dk(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 13th problem.

Our approach to this question (and generalizations) is basedon 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 dk(n) is just the essential dimension of the symmetricgroup Sn. We give results on the essential dimension ofother 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.

Generic polynomials field extensions Galois theory group actions. 


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© Kluwer Academic Publishers 1997

Authors and Affiliations

    • 1
    • 2
  1. 1.Department of MathematicsReed CollegePortlandUSA
  2. 2.Department of MathematicsOregon State UniversityCorvallisUSA

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