Needs["Algebra`FiniteFields`"]

SetFieldFormat[GF[9], 
      FormatType -> FunctionOfCode[GF9]]

Table[
        ElementToPolynomial[GF9[j]*GF9[k], α],  {j, 0, 8}, {k, 0, 
          8}] /. ElementToPolynomial[0, α] -> 0

FieldIrreducible[GF[9], x]

IrreduciblePolynomial[x, 3, 2]

PolynomialMod[(2  α + 1) (α + 
            2), (α^2) + α + 2, Modulus -> 3]