ECTS
5 credits
Component
Faculty of Science
Description
To deepen the basic notions of group and ring theories seen in the previous semester.
Objectives
This EU will address the following:
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 group. The special case of abelian groups.
- Group extensions and semi-direct product. Examples. The special case of vector spaces.
Theory of the rings
- Reminders on ideals, quotient of a ring by an ideal. Theorems of isomorphism and factorization. Ideals of a quotient. Application of the quotient to the construction of extensions of bodies and (small) finite bodies.
- Prime and maximal ideals. Operations on ideals. The Chinese remainder theorem.
Necessary pre-requisites
The algebra courses of L1, L2 and the first semester of L3.
Recommended prerequisites: first semester of L3
Additional information
Hourly volumes:
CM : 22,5
TD : 22,5
TP: -
Land: -