Why is $\mathbb Q(\sqrt 2+\sqrt 3)=\mathbb Q(\sqrt2,\sqrt 3)$?

Question

Why is $\mathbb Q(\sqrt 2+\sqrt 3)=\mathbb Q(\sqrt2,\sqrt 3)$ ?

I am Having problems understanding why this is true.

Any input would be greatly appreciated!

Answer 1

Note that $(\sqrt3+\sqrt2\,)(\sqrt3-\sqrt2\,)=1$.

Thus we call $\xi=\sqrt3+\sqrt2$ and note that $\sqrt3=\frac12\left(\xi+\frac1\xi\right)$and $\sqrt2=\frac12\left(\xi-\frac1\xi\right)$.