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Suppose $F$ is an infinite field, how we could show by compactness that we can find an elementary extension $K$ of $F$ such that $K$ contains a transcendental element $t$?

I only know that if we take a finite subfield from $F$ we could produce an element which is transcendental element over that subfield. but I dont know what is our theory which we should apply compactness on it.

Would be thankful of your help.

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Like most proofs by compactness, this just comes down to carefully writing down what you want. In this case, we want two things:

  1. $K$ is an elementary extension of $F$.
  2. $K$ contains an element $t$ which is transcendental over $F$.

For 1, Given a structure $F$, there is a standard theory $T_F$ such that the models of $T_F$ are exactly the elementary extensions of $F$ (up to isomorphism). What is it?

For 2, we want to add an element satisfying certain properties. So we add a new constant symbol $t$ and write down the properties. What are the properties? $t$ is transcendental over $F$. Easy as pie! Just write down $\lnot p(t) = 0$ for every non-zero polynomial $p\in F[x]$ (this is in the language with a constant symbol for every element of $F$). This gives us a theory $T_{\mathrm{tr}}$.

Now you want to show that $T_F\cup T_{\mathrm{tr}}$ is consistent by compactness. Remember that our field is infinite, and non-zero polynomials have only finitely many zeros...

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