isomorphism between field extension Assume F is a field, E/F and H/F are two field extension. If E is isomorphic to H, then whether exist a isomorphism $\varphi:E \rightarrow H$ such that $\varphi|_F = Id_F$? I think it's worry but I can find some counterexamples. If F = Q, then it is true obviously, but other field? I think it maybe true if E/F is finite extension. Maybe someone give me some references or hints, thanks.

Let $F=\mathbb{Q}(X)$, $E=F, H=\mathbb{Q}(Y)$. Now we consider $F\subset H$ by $X\mapsto Y^2$.
Then $E$ and $H$ is isomorphic as fields by $\mathbb{Q}(X)\to \mathbb{Q}(Y)$ defined by $X\mapsto Y$. (restriction on $F$  of this map is not id$_F$)
And there is no isomorphism $\phi:E\to H$ s.t. $\phi|_F=$id$_F$.
In fact if $\phi|_F=$id$_F$, then $\phi(X)=Y^2$ and $Y$ is not in Im$\phi$. Hence $\phi$ is not isomorphic.
