Prove that $\sqrt{\frac{1+\sqrt5}2}$ is not in $\mathbb Q(\sqrt5)$ Prove that
$$\sqrt{\frac{1+\sqrt5}2} \not \in \mathbb Q(\sqrt5)$$
I think I should start taking the next equation $\sqrt{\frac{1+\sqrt5}2}=p+q\sqrt5$ but I don't see how to continue from here

Let $K = \mathbb Q(\frac{1+\sqrt{5}}{2})$. Note that $N_{K/\mathbb Q}(\frac{1+\sqrt{5}}{2}) = -1$. If $\alpha = \sqrt{\frac{1+\sqrt{5}}{2}} \in K$, then
$$-1 =  N_{K/\mathbb Q}(\alpha^2) = N_{K/\mathbb Q}(\alpha)^2.$$
Since $N_{K/\mathbb{Q}}(\alpha) \in \mathbb Q$, this is a contradiction.
