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.

External Links

Field (Wolfram MathWorld)
Finite Field (Wolfram MathWorld)
Irreducible Polynomial (Wolfram MathWorld)

Permanent Citation

Ed Pegg Jr
​
​"Finite Field Tables"​
​http://demonstrations.wolfram.com/FiniteFieldTables/​
​Wolfram Demonstrations Project​
​Published: March 7, 2011