Why $\mathbb Q(\sqrt 3,\sqrt 5)=\mathbb Q(\sqrt 3+\sqrt 5)$?

Question

I'm trying to show that $\mathbb Q(\sqrt 2+\sqrt 5)/\mathbb Q$ is an extension of degree 4. I know how to prove that $\mathbb Q(\sqrt 2,\sqrt 5)/\mathbb Q$ has degree $4$, and now I want to find an isomorphism between $\mathbb Q(\sqrt 2,\sqrt 5)$ and $\mathbb Q(\sqrt 3+\sqrt 5)$, but didn't work... How can I proceed ?

Answer 1

No isomorphism is needed :

$$(\sqrt 2-\sqrt 5)(\sqrt 2+\sqrt 5)=-3,$$ i.e. $\sqrt 2-\sqrt 5\in \mathbb Q(\sqrt 2+\sqrt 5)$ and

$$\sqrt 2=\frac{\sqrt 2+\sqrt 5+(\sqrt 2-\sqrt 5)}{2},$$ $$\sqrt 5=\frac{\sqrt 2+\sqrt 5-(\sqrt 2-\sqrt 5)}{2}.$$