Topics in algebra and topology: Galois theory 2026

Current affairs October 9

Contents

This is an introduction to Galois theory. We assume the basics of the theory of rings and fields and of the theory of groups as presented in Algebra 1. In particular, we assume that the participants are reasonably comfortable with the construction of fields as quotient rings of polynomials.

We begin by a discussion of field extensions, in particular, splitting fields of polynomials, that are minimal field extensions where a given polynomial splits as the product of first order factors. We discuss automorphisms of field extensions, and introduce an important class of field extensions called Galois extensions, their groups of automorphisms are called Galois groups. We discuss the Galois correspondence that connects the intermediate fields of a Galois extension with the subgroups of its Galois group. We close the course by a discussion of what it means that a polynomial is solvable by radicals, and show that insoluble polynomials of degree five (and higher) with rational coefficients exist. We will develop the required results on group actions and solvable groups.

Course material

There will be a text in Finnish that will be posted here and updated as the course progresses. The material contains exercises, some of which will be done in the course.

Exercises

Contact information

Jouni Parkkonen
Matematiikan ja tilastotieteen laitos
PL 35
40014 Jyväskylän yliopisto