Groups and rings 1

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

This EU will address the following points:

  Group theory

    - Concept of groups, subgroups, and group morphisms. Product of groups. Examples.

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

    - Study ofZ/nZ: Chinese Remainder Theorem, Fermat's Little Theorem, Wilson's Theorem. Generators and subgroups ofZ/nZ, Euler's indicator, Euler's Theorem.

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

  Ring theory

    - Concept of a ring, integral ring, field. Product of rings. Group of inverses of a ring. Algebras over a field. Examples.

    - Subring, subring generated by a part. Ring morphisms. Field 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, GCD, LCM. Principal rings, Euclidean rings, factorial rings.

    - Gauss's lemma and the heredity of factoriality

The algebra courses in L1 and L2, in particular:

- HAX303X Arithmetic of Polynomials

   

Recommended prerequisites: L2 maths

Hourly volumes:

            CM: 27

            TD: 27

            TP: -

            Land: -