Splitting field of $x^2+[1]$ over $\mathbb Z_2$

Question

While finding splitting field of $x^2+[1]$ over $\mathbb Z_2$ I found that all roots of $x^2+[1]$ lies in $\mathbb Z_2$ than how can I find splitting field ?

Answer 1

(Answer given in comments)

$x^2+1 = (x+1)^2$, so it actually already splits. Thus $\Bbb{Z}_2$ is in fact the splitting field. Remember that the $K$ is a splitting field of $f(x) \in F[x]$ if $f(x)$ splits completely over $K$ and if $K = F(\alpha_1,\cdots,\alpha_n)$ where $\alpha_1,\cdots,\alpha_n$ are the roots of $f(x)$ (We then refer to the splitting field since any two splitting fields are isomorphic). In this case, the splitting field would be $\Bbb{Z}_2(1)=\Bbb{Z}_2$.