MATH-317 / 5 crédits

Enseignant: Michel Philippe

Langue: Anglais

## Summary

Galois theory aims at describing the algebraic symmetries of fields. After reviewing the basic material (from the 2nd year course "Ring and Fields") and in particular the Galois correspondence, we will describe applications to classical problems as well as more advanced developments.

## Content

Galois theory aims at describing the algebraic symmetries of fields.

This is a basic topic in mathematics with connections to commutative algebra,  algebraic and arithmetic geometry, number theory and also with more applied areas like cryptology). This is an essential course to anyone interested in the algebra track.

The topics covered may include

• Finiite extensions of fields: separable and normal extensions.
• The Galois group and the Galois correspondence.
• Galois theory of finite fields.
• venerable applications: Ruler and compass construction; equations solvable by radicals: Galois criterion.
• Computation of Galois groups and applications.
• Galois theory of cyclotomic fields.
• Specialization theorems and application to the inverse Galois problem.
• Infinite Galois theory.

## Keywords

Field extension, Galois group

MATH-211

MATH-215

## Important concepts to start the course

Groups, Ring and Fields

## Learning Outcomes

By the end of the course, the student must be able to:

• Quote the results from the course
• Apply the results from the course to other problems
• Prove the main theorems of the course

ex-cathedra

## Expected student activities

Attendance to the course and active participation to the exercise sessions

writen exam

## Supervision

 Assistants Yes Forum No Others moodle page

## Bibliography

Chambert-Loir: A field guide to algebra

James Milne: Galois Theory

## Notes/Handbook

a pdf (in french) will be provided during the course

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MATH-417

MATH-429

MATH-482

MATH-489

MATH-494

MATH-535

## Dans les plans d'études

• Semestre: Automne
• Forme de l'examen: Ecrit (session d'hiver)
• Matière examinée: Algebra V - Galois theory
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel

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