I'd like to calculate derivative of polynomial in $GF(9)\equiv GF(3^2)$. For polynomials in fields of characteristic of $2$ it is quite easy, because operations are modulo 2 which means that there are multiplications by 1 or 0. But in GF(3) operations are modulo 3. For example, there is following polynomial in $GF(9)$: $$p(x)=\alpha x^2 + x$$ Derivative of this polynomial is: $$p'(x)=2 \cdot \alpha x + 1$$ Question is now, how to multiply $2$ by $\alpha$ (elements from $GF(3) (0,1,2)$ by elements in $GF(9) (0, 1, \alpha, \alpha^2, \alpha^3, \alpha^4, \alpha^5, \alpha^6, \alpha^7)$)?
Let generator polynomial for GF(9) be $w(x) = x^2 + x + 2$.
Polynomial representation of field elements can be used too.