Field extensions isomorphic to subfields of each other

Question

Suppose $K_1,K_2$ are field extensions (not necessarily finite) of a field $F$ and that $K_1$ is $F$-isomorphic to a subfield of $K_2$ (i.e. there is a field homomorphism $K_1 \to K_2$ which injects $K_1$ into $K_2$ and is the identity on $F$) and $K_2$ is $F$-isomorphic to a subfield of $K_1$. Is this enough to say that $K_1$ and $K_2$ are isomorphic as fields?

Answer 1

No. We can have $K_1$ injecting into $K_2$ and vice versa, yet not having $K_1$ and $K_2$ isomorphic.

As an example, take isogenous but non-isomorphic elliptic curves $E_1$ and $E_2$ defined over $\Bbb Q$. Their function fields $K_1$ and $K_2$ have the property stated.