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.
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