Title of the course: Galois theory of polynomial equations
Instructor: Dr. Remi Jaoui
Institution: CNRS-Université Lyon 1
Dates: 22-28 July 2024
Prerequisites: Basic knowledge of groups (definition, subgroups, quotients) and fields and polynomials (definition of fields, divisibility in k[X], k[X] is  a principal ring)
Level: Undergraduate (year 3)
Abstract: There are well-known formulas to express the two roots of a quadratic polynomial. Similar (although harder to memorise)  formulas exist to express the solutions of any polynomial of degree three or four. In the beginning of the XIXth century, Evariste Galois gave the first rigorous proof that no such formulas (built using the arithmetic operations and radicals) can accurately describe the roots of a general polynomial of degree five or more.  The goal of this course will be to present Galois’s proof using the language of modern algebra. The central technical result today known as the Galois correspondence relates the structure of the subgroups of a finite group (the Galois group of the polynomial) with the structure of the subextensions of a finitely generated field extension (the field of decomposition of the polynomial).
Language: EN