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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?

Michael Hardy
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    Basically all of classical algebraic number theory, for starters, which was developed to answer questions about the good ol' integers. – fkraiem Nov 29 '15 at 04:46

2 Answers2

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One nice application is proving the following theorem:

A rectangle R with side lengths $1$ and $x$ , where $x$ is irrational, cannot be “tiled” by finitely many squares (so that the squares have disjoint interiors and cover all of R ).

The proof can be found here: http://kam.mff.cuni.cz/~matousek/stml-53-matousek-1.pdf, as Miniature 12 (pg. 39).

user293794
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  • Indeed, this is a very nice problem where the use of a Hamel basis is actually natural. However, I should note that one does not require a full Hamel basis, if one is careful to only take the span of the relevant reals. – user21820 May 15 '20 at 05:51
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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.

Michael Hardy
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Brian M. Scott
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