Summary of Chapter 3

Overview of sections

1. The notion of a polynomial
2. Division of polynomials
3. Polynomial functions
4. Factorization
5. Shift registers
6. Exercises

Overview of contents

Polynomials (with coefficients from an arithmetic system like Z, Q, R, C, Z/nZ) form again an arithmetical system (a `polynomial ring') in which one can add, subtract and multiply. To describe a polynomial we introduced concepts like `coefficient', `term' and `monomial'. From polynomials we can make a polynomial function (with the coefficient ring for the domain and codomain). For polynomials with coefficients from a field we discussed (similarly to the integer case)

• division with remainder and a division algorithm
• Euclid's algorithm for determining the gcd
• the extended Euclidean algorithm for writing the gcd of two polynomials as a linear combination of the two polynomials
• irreducible polynomials (analogous to prime numbers)
• factorization (analogous to integer factorization)