# Intermediate Computational Number Theory

### Robert Campbell

### Contents

- Introduction
- Extension Fields
- Integer Algorithms
- Preliminaries

A simple example is the 8 element Galois Field, GF8. We choose
the polynomial `f`(`x`)=`x`^{3}+`x`^{2}+1,
checking that it is irreducible mod 2. Now we construct GF8 by considering
all quadratic polynomials with coefficients mod 2. Thus the field has the
elements GF8 = {0, 1, x, x+1, x^{2}, x^{2}+1,
x^{2}+x, x^{2}+x+1}. Addition is done
the usual way. Multiplication is done by multiplication, then reducing
the result mod `f`(`x`). Try adding and then
multiplying some elements:

Robert Campbell
Last modified: Dec 9, 2000