# Finite Field Tables

Finite Field Tables

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 .

{0,1,a,b,c,…}

(a+b)+c=a+(b+c)

(a×b)×c=a×(b×c)

a+b=b+a,a×b=b×a

a×(b+c)=a×b+a×c

a+0=a,a×1=a

a+(-a)=0,a×=1

-1

a

a≠0

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.

p

p

n

p

GF()

n

p

p

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.