This is a homework question in a course in Groups and Rings.
The exercise is to determine finitely generated subrings of $\mathbb Q$, where such subrings are to include $1$. The course is following the book Abstract Algebra; theory and applications by Judson.
I've tried to work straight from the subring theorem; for example, suppose $R$ is smallest subring to include the rational numbers $r_1, r_2$, then it must also include all powers of $r_1$, all powers of $r_2$, all products of these powers, powers of $r_1 - r_2$ etc. This approach quickly gets out of hand and I fail to see a general pattern that would allow me to determine all such finitely generated subrings.
Is there a slick way of approaching the problem?