Arithmetic of polynomials

In this course, we will introduce an overview of algebraic structures (ring, ideal, body) before tackling the algebra K[X] and defining the arithmetic on polynomials by making parallels with the arithmetic of integers seen in L1. Computational parts on polynomial functions and rational fractions will be treated (explicit factorizations/decompositions).

Overview of algebraic structures :

groups, rings, bodies, algebras with examples from L1

The algebra K[X]:

 definition, operations, degree, Kn[X] (K=Q, R or C).

Arithmetic of K[X]:

divisibility, irreducible polynomials, Euclidean division, Euclidean algorithm. PGCD and PPCM, Bézout's theorem, Gauss' lemma, decomposition into irreducible factors.

Notion of ideal of a ring, Z and K[X] as principal rings, reinterpretation of divisibility, gcd, cfm in terms of ideals.

Polynomial functions :

Recall: roots, multiplicity, derivation, Taylor formula, characterization of the multiplicity of roots.

Split polynomial, root-coefficients relation. D'Alembert-Gauss theorem, decomposition into irreducible factors in R[X] and C[X]. N-th roots of the unit.

Rational fractions :

definition as a body of fractions of K[X]. Degree, integer part, decomposition into simple elements (on R and C)

HAX203X - L1 Arithmetic and Enumeration

Recommended prerequisites: L1 math

Hourly volumes:

            CM : 15

            TD : 15

            TP:

            Terrain: