In previous work, Ohno conjectured, and Nakagawa proved, relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of ‘extra functional equations’ involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. In the present paper, we generalize their result by proving a similar identity relating certain degree-$\ell$ fields to Galois groups $D_{\ell }$ and $F_{\ell }$, respectively, for any odd prime $\ell$; in particular, we give another proof of the Ohno–Nakagawa relation without appealing to binary cubic forms.

Let $f(x) = \Sigma a_ix^i$ 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 nondegenerate Tschirnhaus transformation. We show that the minimal number ${\rm d}_k(n)$ 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 13th 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 ${\rm d}_k(n)$ is just the essential dimension of the symmetric group ${\rm S}_n$. 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.