In one of my references it says, any Automorphism $\sigma$ of $\mathbb{Q}(\sqrt{2},\sqrt{3})$ is determined by its action on generators $\sqrt{2}$ and $\sqrt{3}$. Im really confuse why is the image of $\sqrt{2}$ under $\sigma$ goes only to either $\sqrt{2}$ or $-\sqrt{2}$. Same goes for $\sqrt{3}$, goes only to either $\sqrt{3}$ or $-\sqrt{3}$.
any insight is highly appreciated.
As a counterexample, if $\sigma(\sqrt 2)=\sqrt 3$, then
$2=\sigma(2)=\sigma(\sqrt 2\sqrt2) = \sigma(\sqrt 2)\sigma(\sqrt 2) = \sqrt 3\sqrt 3 =3$,
as $\sigma$ leaves the base field fixed (eq 1).