Example of GF(9)

The field with 9 elements starts with the integers mod 3, forms polynomials with coefficients in the integers mod 3, and then looks at only the remainders of these polynomials when divided by an irreducible (prime) polynomial of degree two in GF(3).

Exercise:
Verify that the polynomial x^2+1 is irreducible by showing that it has no roots in GF(3). That is, plug in 0, 1 and 2 for x and show that these are not roots.

The elements of GF(9) are therefore: 

    0, 1, 2, x, x+1, x+2, 2x, 2x+1, 2x+2
Here are some examples of addition: 
   1+2=0
   (x) + (2x+1) = 1
   (2x+2) + (2x+2) = 2(2x+2) = x+1
Here are some examples of multiplication: 
   2 * 2 = 1
   x * 2 = 2x
   x * x = x^2 = x^2 + 2(x^2+1) = 3x^2 + 2 = 2
   (x+1) * (2x) = 2x^2 + 2x = 2 * 2 + 2x = 2x + 4 = 2x + 1
Here is the complete multiplication table:

2 x x+1 x+2 2x 2x+1 2x+2
2 1 2x 2x+2 2x+1 x x+2 x+1
x 2x 2 x+2 2x+2 1 x+1 2x+1
x+1 2x+2 x+2 2x 1 2x+1 2 x
x+2 2x+1 2x+2 1 x x+1 2x 2
2x x 1 2x+1 x+1 2 2x+2 x+2
2x+1 x+2 x+1 2 2x 2x+2 x 1
2x+2 x+1 2x+1 x 2 x+2 1 2x

Burton Rosenberg
September 1, 2001