ECTS
6 credits
Component
Faculty of Science
Description
Acquire the basic notions of group and ring theories and illustrate them on examples.
Objectives
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
Necessary pre-requisites
The algebra courses of L1 and L2, in particular :
- HAX303X Arithmetic of polynomials
Recommended prerequisites: L2 math
Additional information
Hourly volumes:
CM : 27
TD : 27
TP: -
Land: -