Find a splitting field for $x^4 - x^2 - 2$ over $\Bbb Z_3$

Question

Find a splitting field for $x^4 - x^2 - 2$ over $\Bbb Z_3$. I can see that $i$ (complex number) would be a root in $\Bbb Z_3(i)$, but I'm not sure if this would be the smallest splitting field, would there be a smaller one?

Answer 1

Notice that $$x^4-x^2-2=x^4+2x^2+1=(x^2+1)^2$$ in $\Bbb Z/3\Bbb Z$. Thus the splitting field has to contain $\alpha $ such that $\alpha ^2+1=0$; and if an extension contains such an $\alpha$, then it contains all the roots since the other root is $-\alpha$. Hence $(\Bbb Z/3\Bbb Z)(\alpha)\cong \dfrac{(\Bbb Z/3\Bbb Z)[t]}{(t^2+1)}$ is the splitting field.