I have two quotient rings over $\Bbb F := GF(3)$: $$\Bbb F[x] / (x^3 -x - 1) \qquad \text{and} \qquad \Bbb F[x] / (x^3 -x + 1) .$$
These things I know: Both quotient rings are irreducible, that means they are the same size, so there should exist isomorphism.
I think these two quotient rings should be a field because they have maximum ideals.
I am trying to find function phi to create isomorphism between these quotient rings but not using brute force method. I have tried several ways but they are not "easy" enough. I know that identity element must be mapped to identity element but it didn't help me to find that exact isomorphic function.
Can somebody push me the correct way or give me tip how and where should I look to easily find isomorphism between these quotient rings without necessity to "heavy" computing.
Hint $$(-x)^3 - (-x) - 1 = -(x^3 - x + 1) .$$