Axiomatize purely transcendental field extension

Question

I have been thinking the following question for several days:

Let $\mathcal{L}$ be the first-order language of theory of rings.

Consider the class P to be class of purely transcendental extension of number fields (finite extension of $\mathbb{Q}$). Then my question is

Is it possible to axiomatize the class P in $\mathcal{L}$ ? That is, can we find a recursive set of $\mathcal{L}$-sentences T so that P=Mod(T) ?( T may be an infinite set of sentences)

I guess we can collect irreducible polynomials over number fields to construct the set T.

The motivation of this question is the following question:

Is it possible to axiomatize the rational function fields over number fields? (i.e. axiomatize the class $\{ K(x)| \mbox{K is algebraic number field} \} $).

I have browsed through the book Model Theoretic Algebra written by C.U.Jensen and H.Lenzing but cannot found answers to the questions above.

Any comments, hints, or answers are welcomed ! Thank you.

Answer 1

This is a quick guess. It needs double-checked. (Honestly I can’t comment on stack exchange yet which is very frustrating)

First, I am assuming you mean all purely transcendental extensions of any number field.

The answer is probably no. To axiomatize the class you are describing is the same as being able to describe the class with a computable $\Pi_1$ sentence of $L_{\omega_1 \omega}$. My attempts at doing so just based on the definitions always yield infinite disjunctions, which pushes the complexity up past $\Pi_1$. Unless there is a nice structural characterization of the class you are referring to, I don’t know how to get around this.

An index set construction ( a common technique in computable structure theory) would likely prove that no computable $\Pi_1$ sentence can axiomatize the class. There seems to be a lot of room for coding hard sets into this class.

Answer 2

Here is a much nicer answer, assuming again that the class P is the class of all purely transcendental extensions of all number fields.

You can use compactness (for types) to show that any theory T whose models include extensions of each number field will also have a model which contains an infinite algebraic extension of Q.

No purely transcendental extension of a number field contains an infinite algebraic extension of Q; therefore there is no such theory P whose models are just those.