Polynomial arithmetic

In this course, we introduce an overview of algebraic structures (ring, ideal, body) before tackling K[X] algebra and defining arithmetic on polynomials, drawing parallels with integer arithmetic seen in L1. Calculations on polynomial functions and rational fractions will be covered (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's lemma, decomposition into irreducible factors.

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

Polynomial functions :

Reminders: roots, multiplicity, derivation, Taylor formula, characterization of root multiplicity.

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

Rational fractions :

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

HAX203X - Arithmetic and enumeration from L1

Recommended prerequisites: L1 maths

Hourly volumes :

            CM: 15

            TD : 15

            TP :

            Terrain :