WOLFRAM|DEMONSTRATIONS PROJECT

Finite Field Tables

​
field size
2
3
2
2
5
7
3
2
2
3
11
13
4
2
17
19
23
2
5
3
3
29
5
2
2
7
6
2
4
3
3
5
7
2
5
3
pixels
1
2
3
4
5
6
7
8
9
10
12
14
16
18
20
25
30
35
40
50
60
80
100
200
polynomial
3
x
+x+1
operation
multiplication
addition
3
x
+x+1 over
Z
2
using ×
A field is a set of elements
{0,1,a,b,c,…}
with the four operations of arithmetic satisfying the following properties. associativity:
(a+b)+c=a+(b+c)
,
(a×b)×c=a×(b×c)
, commutativity:
a+b=b+a,a×b=b×a
, distributivity:
a×(b+c)=a×b+a×c
, zero and identity:
a+0=a,a×1=a
, inverses
a+(-a)=0,a×
-1
a
=1
if
a≠0
.
One example of a field is the set of numbers {0,1,2,3,4} modulo 5, and similarly any prime number
p
gives a field, GF(
p
). A Galois field is a finite field with order a prime power
n
p
; these
GF(
n
p
)
are the only finite fields, and can be represented by polynomials with coefficients in GF(
p
) 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.