Algebra V - Galois theory

MATH-317 / 5 credits

Teacher: Michel Philippe

Language: English

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

Learning Prerequisites

Required courses

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

Teaching methods

ex-cathedra

Expected student activities

Attendance to the course and active participation to the exercise sessions

Assessment methods

writen exam

Supervision

Assistants Yes
Forum No
Others moodle page

Resources

Bibliography

Chambert-Loir: A field guide to algebra

James Milne: Galois Theory

Ressources en bibliothèque

Références suggérées par la bibliothèque

Notes/Handbook

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

Moodle Link

Prerequisite for

MATH-328

MATH-417

MATH-429

MATH-482

MATH-489

MATH-494

MATH-535

 

 

In the programs

  • Semester: Fall
  • Exam form: Written (winter session)
  • Subject examined: Algebra V - Galois theory
  • Lecture: 2 Hour(s) per week x 14 weeks
  • Exercises: 2 Hour(s) per week x 14 weeks
  • Type: optional

Reference week

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