Isomorphism between finite fields with polynomials $a^2 + 1$ and $b^2 + b + 2$

Question

Find an isomorphism between finite fields with irreducible polynomials $a^2 + 1$ and $b^2 + b + 2$ respectively over $F$₃.

I have tried a similar process to this answer but ended up with a very complicated isomorphism with lots of square roots, whereas I am looking for a simple one (e.g. $a$ -> $b + 1$).

Answer 1

All you have to do is find all elements $x\in \mathbb{F}_3(b)$ such that $x^2+1=0$. Since $\mathbb{F}_3(b)$ is of dimension $2$ over $\mathbb{F}_3$, we can write $x=pb+q$ where $ p,q\in \mathbb{F}_3$. Then $$\begin{align} x^2+1&=p^2b^2+2pqb+q^2+1 \\& =p^2(-b-2)+2pqb+q^2+1 \\& =(2p^2+2pq)b+p^2+q^2+1\\&=0.\end{align}$$ Comparing coefficients, we obtain the relation $$\begin{cases} 2p^2+2pq=0 \\ p^2+q^2+1=0 \end{cases}.$$ The first equation is equivalent to $p=0$ or $p+q=0$, and the second $(p,q)=(1,1),(1,2),(2,1),(2,2)$. Their intersection is $(1,2),(2,1)$. So, field homomorphisms $\mathbb{F}_3(a)\to\mathbb{F}_3(b)$ are exactly the two, $a\mapsto 2b+1$ and $a\mapsto b+2$.