Exhibiting an isomorphism between two finite fields
Solution 1:
If $P(x)=x^3-x+1$ and $Q(x)=x^3-x-1$, then $P(-x)=-Q(x)$, so try $$x+\langle x^3-x+1\rangle\mapsto -x+\langle x^3-x-1\rangle.$$
Exhibiting an isomorphism between two finite fields
Solution 1:
If $P(x)=x^3-x+1$ and $Q(x)=x^3-x-1$, then $P(-x)=-Q(x)$, so try $$x+\langle x^3-x+1\rangle\mapsto -x+\langle x^3-x-1\rangle.$$