Let $K \subseteq L$ be a Galois field extension with Gal$(L/K) \cong \mathbb{Z}/4\mathbb{Z}$. Show that $L$ is the splitting field of a polynomial $f(x)=(x^2 −a)^2 −b$ for elements $a,b \in K$ such that $a \neq 0$, $\sqrt{b} \not \in K$ and $\sqrt{a^2−b} \in K$.
I've tried a direct and contrapositive argument, did not lead me anywhere.