Let $K=\mathbb{Q}(\sqrt{p}, \sqrt{q})$ where $p$ and $q$ are distinct primes with $p \equiv q \equiv 3(\bmod 4)$. By computing the relative traces $\operatorname{Tr}_{K / k}(\theta)$ where $k$ runs through the three quadratic subfields of $K$, show that the algebraic integers $\theta$ in $K$ have the form

$$\theta=\frac{1}{2}(a+b \sqrt{p})+\frac{1}{2}(c+d \sqrt{p}) \sqrt{q}$$

where $a, b, c, d$ are rational integers. Show further that if $c$ and $d$ are both even then $a$ and $b$ are both even. Hence prove that an integral basis for $K$ is

$$1, \sqrt{p}, \frac{1+\sqrt{p q}}{2}, \frac{\sqrt{p}+\sqrt{q}}{2} .$$

Calculate the discriminant of $K$.