TITLE: Uses of vector spaces over $\mathbb Q$
QUESTION [8 upvotes]: I know of two applications of vector spaces over $\mathbb Q$ to problems posed by people not specifically interested in vector spaces over $\mathbb Q$:

Hilbert's third problem; and
The Buckingham pi theorem.

What others are there?

REPLY [2 votes]: Assuming $\mathsf{CH}$, one can show that $\Bbb R$ is the union of countably many metrically rigid subsets by viewing it as a vector space over $\Bbb Q$. (A set $D$ in a metric space $\langle X,d\rangle$ is metrically rigid if no two distinct two-point subsets of $D$ are isometric.) This is mentioned in Brian M. Scott and Ralph Jones, Metric rigidity in $E^n$, Proceedings of the American Mathematical Society, Vol. $53$, $1975$, $219{-}222$, though the paper itself contains a different proof of a more general result.