I've a question about my proof of GF(3)[x] for $x^2$+x+2 is a field.
I wanted to prove it as followed: $$ x(1) = 1\\ x(2) = 2\\ x(3) = 0 \to \text{found the 0-Value of the function}\\ x^2+x+2 / (x-3) = x-2 $$ rest = $0 \implies v(x-3) * v^{-1}(x-2) \implies \text{commutative}$
Is it correct like this? Or do I need to prove it differently?
Many thanks in advance!!!