Let $f(x) = x^3 + 2x + 1, g(x) = x^3 + x^2 + x + 2$ be polynomials over $\mathbb{Z}_3$, let $F$ be a splitting field of $f(x)$ and $E$ be a splitting field of $g(x)$. Since $f, g$ are irreducible over $\mathbb{Z}_3$, and their degrees are $3$, degrees of their splitting fields are also $3$.
In my book I have a theorem that says the following (I think it is the one to use): Let $i: K \rightarrow K'$ be a field isomorphism. Let $T$ be a splitting field for $f$ over $K$, $T'$ a splitting field for $i(f)$ over $K'$. Then there is an isomorphism $J: T \rightarrow T'$ such that $j\restriction{K} = i$.
So if I were able to find an isomoprhism $i$ I could tell they are isomorphic. But I don't even know if they are isomoprhic (actually I think they are not).