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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

### Prerequisites

L3 algebra course.

### In brief

**Period :**Semester 7

**Credits :**9

**Number of hours Lectures (CM) :**26h

**Number of hours Tutorial (TD) :**45.5h

**Culmination Code (APOGEE) :**GBMG7U01

**Hing methods :**In person

**Location(s) :**Grenoble - University campus

**Language(s) :**French

### Contacts

### International students

**Open to exchange students**