Showing a polynomial is irreducible

Question

We define $K:=\mathbb{Q}[\sqrt{-3}]$, in particular $e^{2\pi i/3} \in K$.

If $f \in \mathbb{Q}[X]$ is a monic, irreducible polynomial with $\text{deg}(f)=3$, why is $f$ also irreducible over $K$?

Answer 1

Hint: If $f$ were reducible over $K$, $K$ would contain an element of degree $3$ over $\Bbb Q$. Is there such an element?