# Fiches de cours

## Algebra

Lachowska Anna

English

#### 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 rings, factorial rings. Quotient rings. The Chinese remainder theorem.

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

- Fields. Algebraic extensions. 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, ring, ideal, principal ideal, prime ideal, maximal ideal, unique factorization, Euler's totient function, principal ring, factorial ring, 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

Written exam

#### 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.

### Dans les plans d'études

• Semestre
Automne
• Forme de l'examen
Ecrit
• Crédits
3
• Matière examinée
Algebra
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
1 Heure(s) hebdo x 14 semaines
• Passerelle HES - SC, 2019-2020, Semestre automne
• Semestre
Automne
• Forme de l'examen
Ecrit
• Crédits
3
• Matière examinée
Algebra
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
1 Heure(s) hebdo x 14 semaines
• Semestre
Automne
• Forme de l'examen
Ecrit
• Crédits
3
• Matière examinée
Algebra
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
1 Heure(s) hebdo x 14 semaines
• Semestre
Automne
• Forme de l'examen
Ecrit
• Crédits
3
• Matière examinée
Algebra
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
1 Heure(s) hebdo x 14 semaines
• Semestre
Automne
• Forme de l'examen
Ecrit
• Crédits
3
• Matière examinée
Algebra
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
1 Heure(s) hebdo x 14 semaines

### Semaine de référence

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