$E$ is a splitting field of $f(x)$

Question

Let $f(x)=x^2-2 \in \mathbb{Z}_5[x]$.

$f(x)$ is irreducible.

Let $\xi$ be a solution of $f(x)$ in an extension of $\mathbb{Z}_5$.

How can I show that $E=\mathbb{Z}_5(\xi)$ is a splitting field of $f(x)$ ??

Answer 1

Since the two roots are $\xi$ and $-\xi$, you have that the splitting field is $\mathbb{Z}_5(\xi, -\xi) = \mathbb{Z}_5(\xi)$.