G.N. Chebotarev's "On the Problem of Resolvents"

TL;DR

Chebotarev's paper explores the solvability of degree 21 polynomials using algebraic functions, extending Wiman's method, but relies on unproven assumptions about affine space intersections.

Contribution

It extends Wiman's method to show degree 21 polynomials can be solved with algebraic functions of 15 variables, highlighting limitations in existing proofs.

Findings

Degree 21 polynomials can be solved with algebraic functions of 15 variables

Relies on assumptions about intersections in affine space without proof

Extends Wiman's method to higher-degree polynomials

Abstract

This is an English translation of G.N. Chebotarev's classical paper "On the Problem of Resolvents," which was originally written in Russian and published in Vol. 114, No. 2 of the Scientific Proceedings of the V.I. Ulyanov-Lenin Kazan State University. In this paper, Chebotarev extends the method in Wiman's "On the Application of Tschirnhaus Transformations to the Reduction of Algebraic Equations" to argue that the general polynomial of degree 21 admits a solution using algebraic functions of at most 15 variables. However, his and Wiman's proofs assume that certain intersections in affine space are generic without proof.