Field elementary equivalent to $\mathbb{Q}$ I am interested in the theory of rational numbers in the language of rings $\{+,\cdot,0,1\}$.
Of course, this theory is extremely complicated, but I hope to find some non-standard models.
One obvious construction are ultrapowers of $\mathbb{Q}$.
It seems to me that other examples are very hard to get. Some easy necessary properties are characteristic 0, admitting a unique total order, every element not in the prime field being transcendental over it etc.
I would be very thankful for any ideas. Thank you in advance!
 A: It depends on what you mean by "finding" a nonstandard model.
The structures $\mathbb{Q}$ and $\mathbb{N}$ are bi-interpretable. In one direction, $\mathbb{N}$ is definable in $\mathbb{Q}$ by a famous theorem of Julia Robinson. In the other direction, the usual constructions of $\mathbb{Z}$ from $\mathbb{N}$ and $\mathbb{Q}$ from $\mathbb{Z}$ are first-order interpretations.
So a field elementarily equivalent to $\mathbb{Q}$ is essentially the same thing as a model of true arithmetic, the complete theory of $\mathbb{N}$.
There is an extensive literature on nonstandard models of arithmetic (see Kaye's book, for example), but these objects are notoriously complicated. For example, by Tennenbaum's Theorem, no such model is computable. This suggests that we're not going to find any very explicit nonstandard model of the complete theory of $\mathbb{Q}$. Someone else might be able to say something more precise along these lines.
Nontrivial ultraproducts are not very explicit, of course, since they require fixing a non-principal ultrafilter. But if you view ultrproducts as a satisfying construction, I'll point out that model theory gives many other tools for building models of general first-order theories: the compactness theorem, realizing and omitting types, Löwenheim-Skolem, saturated models, Ehrenfeucht-Mostowski models, etc. etc.
