UE Algebra 1

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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.