Chapter 4
Arithmetic modulo polynomials


  1. Congruence modulo a polynomial

  2. The residue class ring

  3. Two special cases

  4. Inverses and fields

  5. Finite fields

  6. Error-correcting codes

  7. Exercises

  8. Summary of Chapter 4

One step beyond arithmetic modulo an integer, is arithmetic `modulo a polynomial' (or several polynomials). Here polynomials that differ by multiples of a fixed polynomial are considered to be equivalent. This construction gives us arithmetical systems that are important in, for example, coding theory and cryptology. In this chapter, R is always one of the sets Z, Q, R, C, Z/nZ (with the usual addition and multiplication), unless explicitly stated otherwise.

Section 4.1
Congruence modulo a polynomial
  1. The notion
  2. Residue class
  3. Class representative
Section 4.2
The residue class ring
  1. Construction
  2. Arithmetical rules
  3. Constants modulo a polynomial
  4. Class rings as vector spaces
  5. Projection onto the residue class ring
Section 4.3
Two special cases
  1. Approximations
  2. Modulo n
Section 4.4
Inverses and fields
  1. Inverses
  2. Fields
Section 4.5
Finite fields
  1. Finite fields
  2. Properties
Section 4.6
Error correcting codes
  1. Coding theory
  2. Linear codes
  3. Construction
  4. Decoding for cyclic codes
  5. BCH bound for cyclic codes