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