Hilbert's 13th problem for algebraic groups

  • Zinovy Reichstein

    University of British Columbia, Vancouver, Canada
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Abstract

The algebraic form of Hilbert's 13th problem asks for the resolvent degree of the general polynomial of degree , where are independent variables. The resolvent degree is the minimal integer such that every root of can be obtained in a finite number of steps, starting with and adjoining algebraic functions in variables at each step. Recently Farb and Wolfson defined the resolvent degree for every finite group and any base field of characteristic . In this setting , where denotes the symmetric group. In this paper we extend their definition of to an arbitrary algebraic {group} over an arbitrary field . We investigate the dependency of this quantity on and show that for any field and any connected group . The question whether can be bigger than for any field and any algebraic group over (not necessarily connected) remains open.

Cite this article

Zinovy Reichstein, Hilbert's 13th problem for algebraic groups. Enseign. Math. (2024), published online first

DOI 10.4171/LEM/1075