Determine the group $\text{Aut}(\Bbb{Q}(\sqrt{2})$,where $\Bbb{Q}(\sqrt{2})=\left\{a+b\sqrt{2}:a,b\in \Bbb{Q}\right\}$.
First, identity and conjugation which is $a+ib\mapsto a-bi$. But I don't know whether there are some other elements in $\text{Aut}(\Bbb{Q}(\sqrt{2}))$. If we can prove that there is no other element, then $\text{Aut}(\Bbb{Q}(\sqrt{2}))\cong \Bbb{Z}_{2}$.