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.


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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