A theory is categorical if it has a unique model up to isomorphism. First-order Peano arithmetic is not categorical, but second-order Peano arithmetic is categorical, with the natural numbers as its unique model. The first-order theory of real closed fields is not categorical, but the second-order theory of Dedekind-complete ordered fields is categorical, with the real numbers as its unique model. ZFC is not categorical, but Morse-Kelley Set Theory with an appropriate axiom about inaccessible cardinals is categorical.
My question is, what theory of the field of rational numbers is categorical? Clearly we can’t characterize $\mathbb{Q}$ as the unique countable ordered field whose order is a dense order without endpoints, because the field of algebraic real numbers also satisfies all that. So what else is required?
Take the first-order theory of fields of characteristic zero (in the language of rings) and add to it the following second-order axiom:
The only field of characteristic zero with this property is $\mathbb{Q}$, so we have the desired categorical theory of rational numbers.