Example of two non-isomorphic fields which embed inside each other

Question

Can you find an example of non-isomorphic fields which embed inside each other?

Most probably we can't but I am looking for extraordinary answer...

Answer 1

A start: Let the two fields be the complex numbers $\mathbb{C}$ and the field $\mathbb{C}(X)$ of rational functions with complex coefficients.

These are non-isomorphic, since one is algebraically closed and the other isn't.

Embedding in one direction is trivial. In the other direction, take a transcendence base for $\mathbb{C}$ and map $X$ to an element of the base, and map the remaining elements of the transcendence base appropriately.

Remark: The "construction" of the embedding uses the Axiom of Choice. My shortcut approach would be to use the fact that the theory of algebraically closed fields of characteristic $0$ is $\kappa$-categorical for every uncountable $\kappa$.

It would be interesting to know whether one can produce an example without using AC.

Answer 2

This is not something I understand well, but I believe you can construct examples as follows.

Let $E$ be an elliptic curve, and let $E'$ be another elliptic curve that is isogenous to $E$, but not isomorphic to $E$.

Then the isogeny should provide an embedding $k(E') \to k(E)$, and its dual should provide an embedding $k(E) \to k(E')$.