Question on the discriminant of a irreducible polynomial

Question

Let $\mathbb{K}_5$ be the field with five elements. I don't know to describe the splitting field of $f(x) = x^{3} + x + 1\in\mathbb{K}[5]$ in terms of a root and the discriminant. Clearly, $f(x)$ is irreducible in $\mathbb{K}_5$. So far, I know that the field extension $\mathbb{K}[\alpha,\sqrt{\Delta}]$ for a root $\alpha$ and the discriminant $\Delta$ of $f(x)$ is the splitting field of $\mathbb{K}_5$ if $\sqrt{\Delta}\notin\mathbb{K}_5$. On the contrary, $\mathbb{K}_5[\alpha]$ is the splitting field of $\mathbb{K}_5$. How do I determine whether or not $\sqrt{\Delta}\notin\mathbb{K}_5$ in this case?