Polynomial arithmetic

In this course, we will introduce an overview of algebraic structures (rings, ideals, fields) before tackling the algebra K[X] and defining arithmetic on polynomials, drawing parallels with the arithmetic of integers seen in L1. Computational aspects of polynomial functions and rational fractions will be covered (explicit factorizations/decompositions).

Overview of algebraic structures:

groups, rings, fields, 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, Euclid's algorithm. GCD and LCM, Bézout's theorem, Gauss's lemma, decomposition into irreducible factors.

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

Polynomial functions:

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

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

Rational fractions:

definition as the set of fractions of K[X]. Degree, integer part, decomposition into simple elements (over R and C)

HAX203X – Arithmetic and Counting in L1

Recommended prerequisites: L1 math

Hourly volumes:

            CM: 15

            TD: 15

            TP:

            Land: