Does the class of fields admit amalgamation?

Question

I believe the isomorphic extension theorem gives a way to "amalgamate" two embeddings $\iota_1:F \rightarrow E$ and $\iota_2: F \rightarrow K$ into embeddings $\iota_3: E \rightarrow \bar{K}$ and $i:K \rightarrow \bar{K}$ that satisfy the amalgamation property with respect to the initial structure $F$, provided $E$ is an algebraic extension of $F$. But I cannot find any literature concerning whether two general (possibly transcendental) field extensions of a field $F$ amalgamate in such a way.

Answer 1

Yes, this is possible. Let $A$ and $B$ be transcendence bases of $E$ and $K$ over $F$, respectively. (These could be $\emptyset$ if the extensions are algebraic.) Assume without loss of generality that $|A|\leqslant |B|$, and let $\alpha:A\to B$ be any injection. Now let $F'=\iota_1(F)(A)$. Then $F'$ is naturally a subfield of $E$, and by definition $E$ is an algebraic extension of $F'$; let $\tau_1:F'\hookrightarrow E$ be the inclusion map. We also have a field morphism $\tau_2:F'\to K$, uniquely determined by declaring its action on $\iota_1(F)\subseteq F'$ to be $\iota_2\circ\iota_1^{-1}$ and its action on $A$ to be given by $\alpha$. Now, since $E$ is algebraic over $F'$, by the isomorphism extension theorem there exists a field $L$ and morphisms $\sigma_1:E\to L$ and $\sigma_2:K\to L$ over $F'$, and now precomposing with $\iota_1$ and $\iota_2$ we are done:

$$\require{AMScd} \require{cancel} \def\diagdownarrow#1{\smash{\raise.6em\rlap{\ \ \scriptstyle #1} \lower.6em{\cancelto{}{\Space{2em}{1.7em}{0px}}}}} \begin{CD} E @>\sigma_1>>L\\ @A\tau_1AA @AA\sigma_2A\\ F' @>\tau_2>> K\\ @A\iota_1AA \diagdownarrow{\iota_2} \\ F \end{CD}$$ The idea here is that we have replaced a question about the possibly non-algebraic extension $E:F$ with one about an algebraic extension $E:F'$, while not including any new elements in $F'$ that are algebraic over $F$ (and thus ensuring that we still have a morphism $F'\to K$ over $F$). This is made possible for field extensions by the machinery of transcendence bases, and it generalizes very nicely to broader contexts via the Noether normalization lemma.