Let's say I have the polynomial $f(x)=x^4 +1 \in \mathbb{Z}_{3}[x]$. I know that this polynomial is reducible in $\mathbb{Z}_{3}[x]$ since it can be decomposed as follows: $$ f(x) = (x^2 +x +2)(x^2 +2x +2) $$ where both factors are irreducible since neither of them have roots in $\mathbb{Z}_{3}$.
My goal is to find two nonisomorphic extension fields in which $f(x)$ has roots. Here's what I did:
The first polynomial has roots $\frac{-1 \pm \sqrt{7}}{2}i$ and the second has roots $-1 \pm i$. I created these field extensions:
$\mathbb{Z}_{3}[\frac{-1 + \sqrt{7}}{2}i]$ and $\mathbb{Z}_{3}[-1+i]$
Now if I did everything right, $f(x)$ has roots over these fields. My worry is showing that these are not isomorphic. I would have to show that there does not exist an isomorphism between the two, but this seems less straight forward than showing that there does exist one.
PS, would it be any different if I had $\mathbb{Z}_{3}[i]$ instead of $\mathbb{Z}_{3}[\frac{-1 + \sqrt{7}}{2}i]$?
Your polynomials have coefficients in $\mathbb{Z}_3$, not in the complex numbers.
However, if you consider the splitting field of $x^2+x+2$ (by adding a root $\alpha$ of it) and the splitting field of $x^2+2x+2$ (by adding a root $\beta$), you get isomorphic fields, because both are fields with $3^2=9$ elements.
The field with $3^4=81$ elements contains the nine element field as a subfield and so $x^4+1$ has roots in it.