isomorphic quotient rings?

Question

I have trouble in determining, whether two rings are isomorphic:

Let's have $R = GF(3)$ and rings $R[x]/(x^2+x+2)$ and $R[x]/(x^2+2x+2)$.

How can one determine whether these two rings are isomorphic? I know that both of them will consist of polynomials with degree 1. And if they are isomorphic, what is the isomorphism?

Appreciate any ideas. Many thanks

Answer 1

Since both $\;x^2+x+2\;,\;\;x^2+2x+2\in\Bbb F_3[x]\;,\;\;\Bbb F_3=GF(3)\;$ are irreducible quadratics, both

ideals $\;I_1:=\langle x^2+x+2\rangle\;,\;\;I_2:=\langle x^2+2x+2\rangle\le\Bbb F_3[x]\;$ are maximal, and thus both quotient

rings $\;\Bbb F_3[x]/I_1\;,\;\;\Bbb F_3[x]/I_2\;$are fields of order $\;3^2=9\;$ ...

Answer 2

Hint $\ $ Over GF($3$): $\,f(x) = x^2+x+2\,\Rightarrow\,f(x\!-\!1) = x^2+2x+2.$