Groups and rings 2

To deepen the basic notions of group and ring theories seen in the previous semester.

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.

The algebra courses of L1, L2 and the first semester of L3.

Recommended prerequisites: first semester of L3

Hourly volumes:

            CM : 22,5

            TD : 22,5

            TP: -

            Land: -