A field is a set of elements with the four operations of arithmetic satisfying the following properties. associativity: , , commutativity: , distributivity: , zero and identity: , inverses if .

One example of a field is the set of numbers {0,1,2,3,4} modulo 5, and similarly any prime number gives a field, GF(). A Galois field is a finite field with order a prime power ; these are the only finite fields, and can be represented by polynomials with coefficients in GF() reduced modulo some polynomial.

In this Demonstration, pick a prime and polynomial, and the corresponding addition and multiplication tables within that finite field will be shown. Squares colored by grayscale represent the fiield elements.