TITLE: A question on morphisms of fields
QUESTION [5 upvotes]: 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?

REPLY [6 votes]: 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.