Proposition 8

(i) If two roots are proportional, the factor of proportionality can only be $\pm 1, \pm \frac{1}{2}, \pm 2$.

(ii) If $\alpha$ and $\beta$ are two non-proportional roots, and if $\| \alpha \| \leq \| \beta \|$, then $n(\alpha, \beta)$ takes one of the values 0, 1, -1.

If a root $\alpha \in \mathbf{R}$ is such that $\frac{1}{2}\alpha \notin \mathbf{R}$, then $\alpha$ is called an indivisible root.