Summary of Chapter 4

Overview of sections

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
   

Overview of contents

Analogous to the arithmetic modullo n in Z, we have the arithmetic modulo a polynomial d in a polynomial ring R[X]. This leads to new arithmetical systems. The topics discussed in this chapter are:

• congruence modulo a polynomial, residue classes;
• the construction of the residue class ring R[X]/(d);
• addition, subtraction (opposite), multiplication in R[X]/(d);
• relation between R[X]/(Xⁿ) and computing with approximations 'upto order n' from analysis.

When R is a field, further topics are:

• invertibility in R[X]/(d);
• the role of Euclid's algorithm;
• the R-vector space structure of R[X]/(d);
• multiplication by an element from this ring defines a linear map;
• R[X]/(d) is a field if and only if d is an irreducible polynomial;
• construction of finite fields.

As an application of the construction of finite fields, error-correcting codes are discussed; these codes are used for safe transport of data over (electronic) communication lines.