Past exam of the options II course of the University of Oxford / 2004 / B / B2 / a

The roots are $\pm \sqrt{-5}$, so one root already supplies both. The polynomial has no rational root and is irreducible of degree two. Thus its splitting field is $\mathbb{Q}(\sqrt{-5})$, with $\boxed{[K:\mathbb{Q}]=2}$.