Notice that, $\mathbb{Z}_5[x]$/$<x^2+x+1>$ and $\mathbb{Z}_5[x]$/$<x^2+2>$ are both finite field with the same 25 elements, thus must be isomorphic. Yes, there is general theorem stated that every finite field with the same order $p^n$ where $p$ is prime, then they are isomorphic. But i want the specific isomorphism between those of two.
Let $\mathbb{Z}_5[x]$/$<x^2+x+1>=\mathbb{Z}_5[\alpha]$ where $\alpha$ is an abstract root of $x^2+x+1$ and $\mathbb{Z}_5[x]$/$<x^2+2> =\mathbb{Z}_5[\beta]$ where $\beta$ is an abstract root of $x^2+2$.
Then, using the step in here, answer by Arturo Magidin. I got $i : \mathbb{Z}_5[\alpha] \rightarrow \mathbb{Z}_5[\beta]$ where $i(\alpha)=\beta+1$.
Now, what im questioning is, how to show that $i$ satisfies homomorphism? And, i think i got problem to show cause im not understanding well what does an abstract root that i mentioned above.
Thus, once i understand to show the homomorphism, the rest is to show the kernel is trivial, hence it is injective, since both size is 25, then they must be surjective too.