I'm working on a comparison between a set theoretical and an axiomatic construction of the hyperrational numbers $^*\mathbb Q$.
So far I have only found the construction of $^*\mathbb Q$ by using rational sequences on ultrafilters. But in the past some people explained to me that the hyperrationals can also be considered as a field extension by adjoining an "infinite" element $\omega$ to the field of rational numbers. One axiom, which is a property of that new element, would be $$\forall q \in \mathbb Q: \ \omega > q.$$
But this can't give us a full description of the field, i.e. there must be more axioms also telling us how the order relation $<$ is extended on $^*\mathbb Q$. In particular $^*\mathbb Q$ is uncountable while the field extension $\mathbb Q(\omega)$ is countable if we don't add additional axioms.
So is there any axiomatic system which fully describes the hyperrational numbers as a field extension of the rational numbers?
What I am aiming for is some kind of "check list" for the properties of the hyperrational numbers constructed by ultrafilters. That is, I want to show that the hyperrationals introduced by ultrafilters indeed satisfy all the necessary axioms in the same way as the real numbers introduced as equivalence classes of Cauchy sequences satisfy the axioms for a complete ordered field. Also, is there a term describing $^*\mathbb Q$ e.g. as "ordered non-standard field"?