SS2017

Introduction to Galois theory

Galois theory deals with roots of univariate polynomials. Galois tells us that the fact that such roots can be expressed by formulas involving only arithmetical operations and radicals - as in the case of polynomials of degree two - depends on the symmetries of the polynomial. These symmetries are encoded in a group, the Galois group. The aim of the course is to introduce this group, its properties and to prove why we cannot have formulas for the roots of polynomials of degree higher than five.

Language: the course is taught in English.

Prerequisites: basics about groups (in particular, normal subgroups, quotients and isomorphism theorems), rings (in particular, polynomial rings) and fields (in particular, extensions of fields). All these prerequisites are provided by the course "Einfuehrung in die Algebra und Diskrete Mathematik".

Exam: the exam is constituted of two parts, a written and an oral one. The written part is mainly about exercises, and, in order to prepare for that, exercises will be distributed during the semester. You are not required to hand in the exercises.

First meeting: to be announced.

Contents of the course:

  • Before Galois: equations of degree at most four
  • Before Galois: Lagrange and Abel
  • Galois groups and Galois correspondence
  • Splitting fields
  • Normal extensions
  • Separable extensions
  • Solvable groups
  • Criterion of solvability by radicals

Dates of the exams: to be fixed.