Originally Posted by
Deveno no, there exists a unique finite field (up to isomorphism) for every prime POWER. 9 is a prime power (3 squared).
note that in a field F of 9 elements we must have 3 = 1+1+1 = 0 (the characteristic of a finite field F must be prime, and the (additive) subgroup generated by 1 must have order a divisor of 9 (by lagrange), the only prime divisor of 9 is 3).
note further that since F -{0} is a group of order 8, we have x^{8} = 1, for all non-zero x.
hence x^{9} + 2x = x(x^{8} + 2). if x = 0, then this is obviously 0.
if x ≠ 0, then x^{8} = 1, so x(x^{8} + 2) = x(1 + 2) = x(3) = x(0) = 0.