Polynomial Arithmetic

In this course, we will provide an overview of algebraic structures (rings, ideals, fields) before introducing the algebra K[X] and defining polynomial arithmetic, drawing parallels with the arithmetic of integers covered in L1. We will also cover computational aspects of polynomial functions and rational fractions (factorization and explicit expansions).

An 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, factorization into irreducible factors.

The concept of an ideal in a ring; Z and K[X] as principal rings; reinterpretation of divisibility, GCD, and LCM in terms of ideals.

Polynomial functions:

Review: roots, multiplicity, derivation, Taylor's formula, characterization of the multiplicity of roots.

Split polynomial, root-coefficient relation. D’Alembert–Gauss theorem, factorization into irreducible factors in R[X] and C[X]. nth roots of unity.

Rational Fractions:

Definition of the field of fractions of K[X]. Degree, integer part, factorization into prime elements (over R and C)

HAX203X – Arithmetic and Counting in L1

Recommended prerequisites: First-year math

Hourly volumes:

            CM: 15

            TD: 15

            Practical Work:

            Lot: