# Fiches de cours

## Galois theory

#### Enseignant(s) :

Eisenbrand Friedrich

English

#### Summary

This course is in an introduction to Galois theory, which is the study of automorphism groups of field extensions. Galois theory is essential for many fields of mathematics such as number theory, algebraic geometry, topology and many more.

#### Content

Ruler and compass constructions

Algebraic and transcendal numbers

Splitting fields, normaility and separability, soluble and simple groups

Automorphis groups of algebraic extensions and the Galois correspondence

Solution of polynomial equations by radical expressions and impossibility thereof for the quintic

Algorithms for calculating Galois groups

Construction of regular n-gons, theorem of Gauss-Wantzel

#### Keywords

polynomials, fields, algebraic extensions, group, Galois group

#### Learning Prerequisites

##### Required courses

Algèbre linéaire avancée I & II

Anneaux et corps

#### Learning Outcomes

By the end of the course, the student must be able to:
• Demonstrate mastery of the basic elements of Galois Theory
• Describe lattice diagrams of algebraic extensions
• Compute Galois groups of small degree polynomials
• Prove basic theorems in Galois theory

#### Teaching methods

Ex-cathedra lectures and exercises

#### Expected student activities

Independent solution of exercises that are proposed during the course.

#### Assessment methods

Written exam

Dans le cas de l¿art. 3 al. 5 du Règlement de section, l¿enseignant décide de la forme de l¿examen qu¿il communique aux étudiants concernés

#### Supervision

 Office hours Yes Assistants Yes Forum Yes

#### Resources

##### Bibliography

Ian Stewart, Galois Theory, Chapman &Hall

### Dans les plans d'études

• Mathématiques, 2019-2020, Bachelor semestre 5
• Semestre
Automne
• Forme de l'examen
Ecrit
• Crédits
5
• Matière examinée
Galois theory
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
2 Heure(s) hebdo x 14 semaines

LuMaMeJeVe
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
En construction
Cours
Exercice, TP
Projet, autre

### légende

• Semestre d'automne
• Session d'hiver
• Semestre de printemps
• Session d'été
• Cours en français
• Cours en anglais
• Cours en allemand