Title of the course: Non-Solvability of Quintics via Radicals
Instructor: Dr. Uğur Efem
Institution: Dyson Institute of Engineering and Technology
Dates: 17-23 August 2026
Prerequisites: Basic knowledge of groups, rings, fields and (single variable) polynomials over fields. A standard course in linear algebra (knowledge of elementary concepts such as vector space, basis, linear independence, span and dimension is enpugh). Familiarity with field extensions, and group actions will be useful, but I will cover these if necessary.
Level: Graduate, advanced undergraduate, beginning undergraduate (Assuming basic knowledge of groups, fields and linear algebra. Normally, students who completed their second year will know these.)
Abstract: First proof of unsolvability of a quintic polynomial with radicals is often attributed to Abel and Ruffini. However, a Galois theoretic proof of this fact is often presented instead of their original proof. In this short course, the main goal is to present the original proof of Abel and Ruffini. I will explain in detail how they proved this fact without using Galois theory. Although the proof will not use Galois theory, we will use the modern language of field theory. So, we will often talk in terms of splitting fields and minimal polynomials.
I will try to keep the course as self contained as possible. The main ingredients of the proof, namely, symmetric polynomials, radical field extensions, radical towers, symmetric and alternating groups and their actions on the roots of a polynomial will be explained as we progress. If time permits, I will talk about further works of Abel, Galois and Kronecker giving conditions on the solvability of a polynomial using radicals.
Language: TR, EN