I have a question about the uniqueness of splitting field $L$ of polynomial $X^2-2$ over $\mathbb{Q}$. By factorizing $X^2-2 = (X-a)(X-b)$, we know the splitting field is $L = \mathbb{Q}(a,b)$, and we can show $\mathbb{Q}(a,b) = \mathbb{Q}(a) = \mathbb{Q}(b)$. An obvious choice of $a$ would be $\sqrt{2}$.There is a proposition saying here any two splitting fields over $\mathbb{Q}$ of this polynomial are $\mathbb{Q}$-isomorphic, but another splitting field of this polynomial can be $L$ generated by two elements $u,v$ unrelated to $a,b$, so I am wondering if there exists an isomorphism $\phi: \mathbb{Q}(a,b)\rightarrow \mathbb{Q}(u,v)$ which is not the identity map? Or there is only one possibility of $\phi$?
Thank you in advance.
There are exactly two possibilties for $\varphi$ and that is because there are two automorphisms of ${\mathbb Q}(\sqrt2)$, namely the identity and the one that sends $\sqrt2$ to $-\sqrt2$.