Prove that $(1,\frac{1+\sqrt{5}}{2},\frac{1+\sqrt{13}}{2},\frac{1+\sqrt{5}+\sqrt{13}+\sqrt{65}}{4})$ is an integral basis for $K$

Question

Prove that $\left(1,\frac{1+\sqrt{5}}{2},\frac{1+\sqrt{13}}{2},\frac{1+\sqrt{5}+\sqrt{13}+\sqrt{65}}{4}\right)$ is an integral basis for $K=\Bbb{Q}(\sqrt{5},\sqrt{13})$.

What is $d(K)$?

I could not do this question. Can you help me please? Thanks for any help!

Answer 1

Hint: Use the following proposition:

Let $L,L'$ be number fields of degrees $n,n'$ over $\Bbb Q$ such that $L \cap L' = \Bbb Q$. Let $(w_i)_{i=1,\dots,n}$ and $(w'_j)_{j=1,\dots,n'}$ be the integral basis of $L$ and $L'$ respectively. Suppose the discriminants $d,d'$ of $L,L'$ are relatively prime. Then $$(w_iw'_j)_{\substack{i=1,\dots,n \\ j=1,\dots,n'}}$$ is an integral basis for $LL'\mid \Bbb Q$ and the discriminant is given by $d^{n'}d'^n$.