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,…}
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
-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.
External Links
External Links
Permanent Citation
Permanent Citation
Ed Pegg Jr
"Finite Field Tables"
http://demonstrations.wolfram.com/FiniteFieldTables/
Wolfram Demonstrations Project
Published: March 7, 2011