## UE Algebra 1

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.