Groups and Rings 1

Acquire a basic understanding of group and ring theory and illustrate these concepts with examples.

This lesson will cover the following topics:

 Group Theory

    - The concept of groups, subgroups, and group morphisms. Group products. Examples.

    - A subgroup generated by a set, a cyclic subgroup. Order of an element in a group, Lagrange’s theorem, index of a subgroup.

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

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

 Ring Theory

    - The concept of a ring, an integral ring, and a field. Product of rings. The group of invertibles of a ring. Algebras over a field. Examples.

    - Subring, a subring generated by a subset. Ring morphisms. The field of fractions of an integral ring.

    - Characteristics of a ring, Frobenius morphism, the 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 inheritance of factoriality

The algebra courses in the first and second years of undergraduate studies, specifically:

- HAX303X Polynomial Arithmetic

Recommended prerequisites: L2 Math

Hourly volumes:

            CM: 27

            TD: 27

            Practical Work: -

            Land: -