Groups and rings 2

Further explore the basic concepts of group and ring theory covered in the previous semester.

This EU will address 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 group. The special case of abelian groups.

    - Group extensions and semi-direct products. Examples. The special case of vector spaces.

 Ring theory

    - Review of ideals, quotient of a ring by an ideal. Isomorphism and factorization theorems. Ideals of a quotient. Application of the quotient to the construction of field extensions and (small) finite fields.

    - Prime and maximal ideals. Operations on ideals. The Chinese remainder theorem.

Algebra courses in the first year, second year, and first semester of the third year.

Recommended prerequisites: first semester of L3

Hourly volumes:

            CM: 22.5

            TD: 22.5

            TP: -

            Land: -