Relative trace of element

Question

This is a pretty simple question, but I've been stuck on it for ages. Basically I want to find the trace of $$\theta = a+b\sqrt{p}+c\sqrt{q}+d\sqrt{pq}$$ over the relative fields $K = \mathbb{Q}(\sqrt{p},\sqrt{q})$ over $\mathbb{Q}(\sqrt{pq})$. Then clearly $K$ has a basis $1,\sqrt{p}$ over $\mathbb{Q}(\sqrt{pq})$. However I can't figure how to write $\theta$ as a linear combination of $1$ and $\sqrt{p}$ over $\mathbb{Q}(\sqrt{pq})$. Once I get that bit, calculating the trace should be easy.

Answer 1

The conjugate of $\theta$ over $\Bbb Q(\sqrt{pq})$ is $a-b\sqrt p-c\sqrt q +d\sqrt{pq}$ so the trace you seek is $2(a+d\sqrt{pq})$.

Note that $$\theta =a+d\sqrt{pq}+\sqrt p\left(b+\frac cp\sqrt{pq}\right).$$