TITLE: Show that $\frac{3}{5} + i\frac{4}{5}$ isn't a root of unity
QUESTION [5 upvotes]: Intuitively, I see why $\frac{3}{5} + i\frac{4}{5}$ is not a root of unity because $\frac{2\pi}{\arctan(4/3)}$ appears to be irrational when I plug into my calculator. But how to I show this rigorously?I think contradiction should work, but still I wasnt able to show this rigorously. Any help will be appreciated. (Also: I first tried using the idea that roots of unity had angle $\frac{2\pi}{n}$ for n in the natural numbers but then I realized that n should be replaced by rational number because $e^{\frac{i4\pi}{7}}$ is a root of unity too.)

REPLY [2 votes]: Here is another kind of proof. There is perhaps room to explore the hinterland of the concept of an algebraic integer - this is a concept, though, which emerges naturally from this kind of question.
Let $5x=3+4i$ then $(5x-3-4i)(5x-3+4i)=25x^2-30x+25=0$ so that $5x^2-6x+5=0$ is the minimal polynomial for $x$ over the integers. Since it is not monic, $x$ is not an algebraic integer and can't therefore be a root of unity (roots of unity are algebraic integers).