Let $K=\mathbb{Q}(\sqrt{2}, \sqrt{p})$ where $p$ is a prime with $p \equiv 3 \operatorname{(\operatorname {mod}4)\text {.Bycomputingthe}}$ 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

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

where $a, b, c, d$ are rational integers. By further computing the relative norm $\mathrm{N}_{K / k}(\theta)$ where $k=\mathbb{Q}(\sqrt{2})$, show that 4 divides

$$a^{2}+p b^{2}-2\left(c^{2}+p d^{2}\right) \quad \text { and } \quad 2(a b-2 c d)$$

Deduce that $a$ and $b$ are even and $c \equiv d(\bmod 2)$. Hence verify that an integral basis for $K$ is

$$1, \quad \sqrt{2}, \quad \sqrt{p}, \quad \frac{1}{2}(1+\sqrt{p}) \sqrt{2} .$$