Degrees incorporating this pedagocial element :

### Description

**Programme**

Complements on rings

- Factor rings, Gauss theorem
- Algebra of polynomials in a finite number of variables, symmetric polynomials, formal series in one variable
- Link between coefficients and roots of a polynomial
- Irreducible polynomials, irreducibility criteria (Eisenstein, etc.)

Introduction to Modules and Algebras

- Concept of module (on a commutative ring), examples on K [X] and on Z, homomorphisms
- Free modules, counter-examples
- Operations on rows and columns of a matrix with coefficients in a Euclidean ring, invariants of similarity
- Theorem of the base adapted on a Euclidean ring
- Structure of finite abelian groups (uniqueness will be admitted)
- Concept of algebra, first examples (matrices, algebra of a finite group on C)

Body (the bodies considered are commutative)

- Body of fractions of an integral ring, rational fractions
- Roots of the unit, finite subgroups of the group of invertible elements of a body, cyclotomic polynomials
- Degree of body extension, multiplicity
- Rupture field, splitting field
- Algebraically closed body (definition), the body C of complex numbers is algebraically closed
- Algebraic element, transcendental, minimal polynomial, algebraic extension
- Finite bodies, structure, existence and uniqueness

### Recommended prerequisite

L3 algebra course.

### In brief

**Period :**Semestre 7

**Credits :**9

**Number of hours**

- Lectures (CM) : 33h
- Tutorials (TD) : 48h

**Hing methods :**In person

**Location(s) :**Grenoble

**Language(s) :**French

## Contact(s)

**Program director**

Odile Garotta

### International students

**Open to exchange students**