I've found several uses of the phrase "linearly independent over the rationals" that imply that any set of irrationals is linearly independent over the rationals if it is pairwise linearly independent over the rationals, but I can't find a reference to justify that claim. Can someone point me to such a reference?
What I want to show is that given some $k \notin ℚ$ for which $k^n$ is irrational when $n \in ℤ$ and $0, the set $\lbrace k^n, 0 is linearly independent over the rationals. (For example, let $k=p^{1/q}$ for some prime $p$ and integer $q$, then $m=q$.) The reference I can't find would be sufficient to show that; I wonder if I'm failing to find one because it's put in some other terms that I'm failing to recognize as equivalent, and if that's the case I may need to reframe the question to ask for a translation.
linear-algebra
number-theory