ECTS
5 credits
Component
Faculty of Science
Description
Deepen the basic notions of group and ring theory covered in the previous semester.
Objectives
This UE will cover the following points:
Group theory
- Action of a group on a set, quotient of a set by a group action. Cayley's theorem. Class formula, Burnside formula. Application to enumeration
- Sylow's theorems and applications.
- Distinguished subgroup, quotient of groups. Isomorphism and factorization theorems. Simple groups. The special case of abelian groups.
- Group extensions and semi-direct product. Examples. The special case of vector spaces.
Ring theory
- Reminders on ideals, quotient of a ring by an ideal. Isomorphism and factorization theorems. Ideals of a quotient. Application of the quotient to the construction of extensions of finite and (small) finite fields.
- Prime and maximal ideals. Operations on ideals. The Chinese remainder theorem.
Necessary prerequisites
Algebra courses in L1, L2 and the first semester of L3.
Recommended prerequisites: first semester of L3
Further information
Hourly volumes :
CM: 22.5
TD: 22.5
TP: -
Land: -