Groups and rings 1

Acquire the basic notions of group and ring theories and illustrate them on examples.

This EU will address the following:

 Group theory

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

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

    - Study of Z/nZ: Chinese remainder theorem, Fermat's small theorem, Wilson's theorem. Generators and subgroups of Z/nZ, Euler indicator, Euler theorem

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

 Theory of the rings

    - 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. Morphisms of rings. 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 heredity

The algebra courses of L1 and L2, in particular :

- HAX303X Arithmetic of polynomials

Recommended prerequisites: L2 math

Hourly volumes:

            CM : 27

            TD : 27

            TP: -

            Land: -