A necessary condition for solvability by radicals

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.