Let $F$ be a field characteristic not 2 and let $K=F(\sqrt{a}, \sqrt{b})$ be a biquadratic field extension (of degree 4 ) of $F$, for $a, b \in F^{\times}$not squares. Suppose that $b=x^{2}-a y^{2}$ for some $x, y \in F$ (i.e., $b$ is a norm for the quadratic extension $F(\sqrt{a}) / F)$. Prove that there is a field extension $L$ of $K$ that is Galois over $F$ with Galois group the dihedral group of order 8 .
I'm not sure how to approach this. Some guidance would be greatly appreciated!