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.

A rather general way of constructing an example is to take a Galois extension $E/K$ which admits an intermediate field $F$ such that $F/K$ is not Galois. In other words, there is $\varphi \in \operatorname{Gal}(E/K)$ such that $\varphi(F) \ne F$. Now just take $H = E$.
For instance you may take $K = \mathbb{Q}$, $E = H = K(\sqrt[3]{2}, \omega)$, where $\omega$ is a primitive third root of unity. $E/K$ is Galois, as $E$ is the splitting field over $K$ of the polynomial $x^{3} - 2$. Let $F = K(\sqrt[3]{2})$. There is an element $\varphi$ of $\operatorname{Gal}(E/K)$ such that $\varphi(\sqrt[3]{2}) = \omega \sqrt[3]{2}$, so that $\varphi(F) = K(\omega \sqrt[3]{2}) \ne F$.
