# Algebra

## Summary

Study basic concepts of modern algebra: groups, rings, fields.

## Content

- Algebraic structures: sets, groups, rings, fields.

- Groups. Subgroups. Homomorphisms of groups, normal subgroups, quotients. Cyclic groups, symmetric groups. Classification of finite abelian groups.

- Rings. Homomorphisms of rings. Ideals, principal, prime and maximal ideals, principal ideal domains. Quotient rings. The Chinese remainder theorem.

- Examples of rings. Integers. basic properties. Euler's and Fermat's theorems. Polynomial rings. GCD, unique factorization.

- Fields. Finite fields. Characteristic of a field.

## Keywords

Group, homomorphism, subgroup, normal subgroup, quotient group, cyclic group, symmetric group, order of the group, order of an element in the group, finite abelian groups. Ring, ideal, principal ideal, maximal ideal, unique principal ideal domain, Euler's totient function, field, finite field, characteristic of a field.

## Learning Prerequisites

## Required courses

Linear Algebra I, Analyse I

## Recommended courses

Linear Algebra I, Analyse I, Analyse II

## Learning Outcomes

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

- Apply concepts and ideas of the course
- Reason rigorously using the notions of the course
- Choose an appropriate method to solve problems
- Identify the concepts relevant to each problem
- Apply concepts to solve problems similar to the examples shown in the course and in problem sets
- Solve new problems using the ideas of the course
- Implement appropriate methods to investigate the structure of a given group, ring or field, and study their properties

## Teaching methods

Lectures and exercise sessions

## Assessment methods

Three short in-class tests (15% of the grade)

Written exam (85 % of the grade)

## Supervision

Office hours | No |

Assistants | Yes |

Forum | No |

## Resources

## Bibliography

1. D.S. Dummit, R. M. Foote, Abstract Algebra. Wiley, Third Edition

2. S. Lang, Undergraduate Algebra. Undergraduate texts in Mathematics. Springer-Verlag, Inc. New York, second edition, 1990.

3. L. Childs, A Concrete Introduction to Higher Algebra. Undergraduate texts in Mathematics, Springer-Verlag, Inc. New York, 1995.

## Ressources en bibliothèque

- Undergraduate Algebra / Lang
- A Concrete Introduction to Higher Algebra / Childs
- Abstract algebra /Dummit

## Notes/Handbook

Complete lecture notes will be available in PDF

## Moodle Link

## In the programs

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Algebra**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Algebra**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Algebra**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Algebra**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Algebra**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks