A necessary condition for solvability by radicals

TL;DR

This paper discusses a necessary condition for solving polynomial equations by radicals, using Galois theory, and provides an overview suitable for a course setting.

Contribution

It presents a proof of a necessary condition for solvability by radicals based on Galois theory, complementing the upcoming discussion of sufficient conditions.

Findings

Provides a proof of the necessary condition for solvability by radicals

Summarizes key Galois theory results relevant to polynomial solvability

Prepares for a subsequent presentation of sufficient conditions using linear algebra

Abstract

This note was prepared as a handout for the MAT401 course ``Polynomial equations and fields", taught at the University of Toronto in Spring 2026. It presents a proof of a necessary condition for the solvability of algebraic equations by radicals, based on Galois theory. We begin with a brief overview of the relevant basic results from Galois theory, as covered in MAT401, and use -- without proof -- several standard (and relatively simple) results from the course textbook [1]. The sufficient condition for solvability by radicals, which is based on linear algebra, we will present in the next handout.