Suppose $F$ and $K$ are fields both generated by a common subring $D$, which is a domain. My question is, why is there a unique isomorphism between $F$ and $K$ which is the identity on $D$?
Wouldn't the field generated by $D$ be unique? So $F=K$, and then any isomorphism is determined by its images on a generating set, that is, $D$, so such an isomorphism is unique? If this is so, how can we be sure that there exists an isomorphism which restricts to the identity on $D$?
Second thoughts: Let $F$ be the field of fractions of $D$. Then I let $\iota\colon D\to F_1$ be the identity embedding, which is known to have a unique extension to a monomorphism of $F$ into $F_1$. Since $F_1$ is generated by $D$, it is isomorphic to $F$, so this unique monomorphism is an isomorphism? Likewise, there is a unique isomorphism between $F$ and $F_2$ which is the identity on $D$, so there is a unique isomorphism between $F_1$ and $F_2$ which is the isomorphism on $D$. Is this right?