A question on morphisms of fields

Let $A,B$ be two fields. Let $\phi:A\rightarrow B$ and $\psi:B\rightarrow A$ be two morphisms of fields. Can i conclude that $A$ and $B$ are isomorphic fields?

My guess is yes, because every morphism of fields is injective, hence in this case $B$ contains an isomorphic copy of $A$, which in turns contains one copy of $B$. If this is right, how can i formalize it?

Solutions Collecting From Web of "A question on morphisms of fields"

This is an occasion, when instincts developed over finite extensions of (prime) fields lead one astray.

The first counterexamples that come to mind need a bit of background from the theory of elliptic curves. It is quite possible for there to be isogenies going back and forth between two non-isomorphic elliptic curves, $E_1$ and $E_2$. The isogenies give rise to embeddings between the corresponding function fields
$K(E_1)$ and $K(E_2)$ (take for example $K=\mathbb{C}$ to avoid several algebraic pitfalls). Yet, if the two elliptic curves are not isomorphic, the functions fields won’t be isomorphic either.