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So I want to find the isomorphism $\phi$ that takes $F = \mathbb{Z}_3/\langle x^3 - x - 1\rangle$ to $E = \mathbb{Z}_3/\langle x^3 - x + 1\rangle$. I understand that these are both finite fields of size 3^3 since $x^3 - x - 1$ and $x^3 - x + 1$ are both irreducible polynomials of degree 3. I feel like a good start would be to take the subfield $\mathbb{Z_3}$ in $F$ to the subfield $\mathbb{Z_3}$ in $E$. Or maybe find the right generator of each multiplicative group of size $3^3-1$ of each field and send those to each other.

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  • $\begingroup$ You can try to find a pair of matching generators, but IMVHO that is not very natural. Then you are looking for needles from two hay stacks as opposed to just one. I guess the second idea isn't nearly as bad as I made it sound, because any of the conjugates of the matching generator can serve in the role. After all, the isomorphism is not unique. $\endgroup$ – Jyrki Lahtonen Oct 10 '14 at 6:56
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If $P(x)=x^3-x+1$ and $Q(x)=x^3-x-1$, then $P(-x)=-Q(x)$, so try $$x+\langle x^3-x+1\rangle\mapsto -x+\langle x^3-x-1\rangle.$$

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  • $\begingroup$ All the non constant terms have odd degree, and the constants differ in sign only, so this sorta stands out. I have a vague recollection that Dilip Sarwate has solved this one earlier... $\endgroup$ – Jyrki Lahtonen Oct 10 '14 at 6:31
  • $\begingroup$ Not quite. Prof. Sarwate needed the reciprocal. $\endgroup$ – Jyrki Lahtonen Oct 10 '14 at 6:33

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