Hilbert's 13th Problem for Algebraic Groups

Abstract

The algebraic form of Hilbert's 13th Problem asks for the resolvent degree rd(n) of the general polynomial f (x) = x n +a 1 x n−1 +. . .+a n of degree n, where a 1 , . . . , a n are independent variables. The resolvent degree is the minimal integer d such that every root of f (x) can be obtained in a finite number of steps, starting with C(a 1 , . . . , a n ) and adjoining algebraic functions in d variables at each step. Recently Farb and Wolfson defined the resolvent degree rd k (G) of any finite group G and any base field k of characteristic 0. In this setting rd(n) = rd C (S n ), where S n denotes the symmetric group. In this paper we define rd k (G) for every algebraic group G over an arbitrary field k, investigate the dependency of this quantity on k and show that rd k (G) 5 for any field k and any connected group G. The question of whether rd k (G) can be bigger than 1 for any field k and any algebraic group G over k (not necessarily connected) remains open.