• ECTS

    5 credits

  • Component

    Faculty of Science

Description

Deepen the basic notions of group and ring theory covered in the previous semester.

 

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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.

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Necessary prerequisites

Algebra courses in L1, L2 and the first semester of L3.

 

Recommended prerequisites: first semester of L3

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Further information

Hourly volumes :

            CM: 22.5

            TD: 22.5

            TP: -

            Land: -

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