TITLE: Is the set $\{\frac{1}{a\,-\,\pi}\mid a\in\mathbb{Q}\}$ linearly independent over $\mathbb{Q}$?
QUESTION [8 upvotes]: The following problem is from Golan's linear algebra book. I have posted a proposed solution in the answers.
Problem: Consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Is the subset $$\left\{\frac{1}{a-\pi}\;\middle\vert\; a\in\mathbb{Q}\right\}$$ linearly independent?

REPLY [4 votes]: Well, if you are going to assume known that $\pi$ is transcendental over $\mathbb Q$ ... you may as well use any other transcendent instead.  For example, prove that
$$
\frac{1}{a-z},\qquad a \in \mathbb Q
$$
in the vector space of meromorphic functions is linearly independent over $\mathbb Q$.  In fact, more is true: linearly independent over $\mathbb C$.  We can see none of these is a linear combination of some others by comparing their poles.