Groups and rings 1

Acquire the basic notions of group and ring theory and illustrate them with examples.

This UE will cover the following points:

 Group theory

    - Notion of group, subgroup and group morphisms. Product of groups. Examples.

    - Subgroup generated by a part, cyclic subgroup. Order of an element in a group, Lagrange's theorem, index of a subgroup.

    - Study of Z/nZ: Chinese remainder theorem, Fermat's little theorem, Wilson's theorem. Z/nZ generators and subgroups, Euler indicatrix, Euler theorem

    - Study of the dihedral group. Study of symmetrical and alternating groups.

 Ring theory

    - Notion of ring, integral ring, body. Product of rings. Group of inversibles of a ring. Algebras over a body. Examples.

    - Sub-ring, sub-ring generated by a part. Ring morphisms. Body of fractions of an integral ring.

    - Characteristic of a ring, Frobenius morphism, case of finite fields.

    - Ideal of a commutative ring, principal ideal, principal ring

    - Divisibility in integral rings: irreducible and prime elements, PGCD, PPCM. Principal rings, Euclidean rings, factorial rings.

    - Gauss's lemma and factoriality inheritance

L1 and L2 algebra courses, in particular :

- HAX303X Polynomial arithmetic

Recommended prerequisites: L2 maths

Hourly volumes :

            CM: 27

            TD : 27

            TP: -

            Land: -