In what classes of fields does CSB hold? That is to say, in what classes of fields is it true that if there exist embeddings $F\to K$ and $K\to F$ then $F$ and $K$ must be isomorphic?
I know this holds for algebraically closed fields, but all of the counter-examples I've seen are variations on the same idea ($F=\overline{\mathbb Q(x_0,x_1,x_2,...)}$ and $F(x)$).
Does CSB hold for fields of finite transcendence degree? What about for fields with no algebraically closed subfields?
A trivial example of a family that satisfy CSB is the set of completions $\{\mathbb{Q}_{v}\}_{v}$ of $\mathbb{Q}$. They satisfy the condition that you require about not containing any ACF, however their transcendence degree is big. This is a trivial example in the sense that even if one of such fields is embedding in the other then they are isomorphic. More generally if you take all locally compact fields the same should be true.
No. For any field $k$, there exists a pair of isogenous elliptic curves over $k$ that are not isomorphic. The dual isogeny yields suitable maps of function fields that are not isomorphisms. If your base field has finite transcendence degree and does not contain an algebraically closed subfield, then the same is true for the function fields.
Edit: I see that this answer is essentially identical to Robin Chapman's answer to a more general question, and to KConrad's answer to a slightly different question.
