Let $F$, $K$, and $L$, be fields, and suppose we are given maps $F\to K$ and $F\to L$. I claim that there exists some field $\Omega$, and maps $K\to\Omega$ and $L\to\Omega$ such that $F\to K\to\Omega = F\to L\to\Omega$.
Why is this true though?
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Well for algebraic extensions, this amounts to just adjoining a root (or roots) of polynomials, by taking polynomial rings over the field, and modding out by such a polynomial. Of course, you can just do this twice. For more complicated extensions, I don't know. – A. Thomas Yerger Feb 10 '17 at 05:57
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Why do you claim that something is true if you don't know why it is true?! There are lots of verbs available —I guess, I imagine, I think that blah should be true, I expect, I'd love X to hold, it would be useful for me that X be true, and so on, all the way to I conjecture— so there is no need to abuse one of the ways that we have to, well, claim that something is true. – Mariano Suárez-Álvarez Feb 10 '17 at 05:59
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The standard proof of the fact you guessed is George's answer at http://math.stackexchange.com/questions/1098660/field-extensions-without-a-common-extension – Mariano Suárez-Álvarez Feb 10 '17 at 06:07
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The fact that this is true is assumed somewhere in the Stacks project. I just saw no link to a proof anywhere. – Dominic Wynter Feb 10 '17 at 14:24