A question on Automorphism on Galois Theory [duplicate]
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).
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).