Isomorphism map between $\frac{\mathbb{Z}_3[x]}{(x^2+x+2)}$ and $ \frac{\mathbb{Z}_3[x]}{(x^2 + 1)} $

Question

$\frac{\mathbb{F}_3[x]}{(x^2+x+2)} \cong \frac{\mathbb{F}_3[x]}{(x^2 + 1)} $

I know $\frac{\mathbb{F}_3[x]}{(x^2+x+2)} $ and $ \frac{\mathbb{F}_3[x]}{(x^2 + 1)} $ is isomorphic because they are both 2-degree extension of $ \mathbb{F}_3 $ .

But I cannot contract explicit isomorphism between them.

Could you show me isomorphism map between $\frac{\mathbb{F}_3[x]}{(x^2+x+2)}$ and $ \frac{\mathbb{F}_3[x]}{(x^2 + 1)} $?

Thank you for your help.

Answer 1

Try $\phi:\mathbb{F}_3[X]\to\mathbb{F}_3[X]$ defined by $\phi:X\mapsto X+1$ whilst leaving everything in $\mathbb{F}_3$ alone.

This sends $X^2+X+2$ to $X^2+1$, so it sends the ideal $(X^2+X+2)$ to the ideal $(X^2+1)$, so when you quotient out $\mathbb{F}_3[X]$ by $(X^2+X+2)$ or by $(X^2+1)$, you get isomorphic fields.