Consider $f(x) = x^2 + 1$ and $g(x) = x^2 + 4$, two irreducible polynomials in $\mathbb{Z}_3 [x]$. Let $\alpha$ and $\beta$ be a zero of $f(x)$ and $g(x)$, respectively. Are $\mathbb{Z}_3 (\alpha)$ and $\mathbb{Z}_3 (\beta)$ isomorphic?
I think the answer is yes, because any irreducible polynomial in $\mathbb{Z}_3 [x]$ is just each polynomail "shifted" by 3, right? Which means all extension fields would behave similarly, and hence be isomorphic. Is my line of thinking correct? Thank you.