Algebraic Characterizations of Angle Multisections over Rings

Let $n,$ $m\ge 2$ be integers, and let $R$ be a subring of $\mathbb{R}$ with field of fractions $F\mathrm{.}$ In this article, we generalize the rational angle bisection problem previously proposed by the author to the following problem: which linearly independent vectors $\mathit{a},$ $\mathit{b}\in R^n$ form an angle with a sequence of $m$-sector vectors lying in $R^n$? When $\mathit{a}$ and $\mathit{b}$ are nonorthogonal, we prove that this condition is equivalent to the existence of a root in $F$ of a certain $m$-th degree polynomial over $R\mathrm{.}$ In particular, when $R=\mathbb{Z},$ the condition holds if and only if the polynomial has a root among the divisors of its constant term. When $m=2^e$ with an integer $e\ge 1,$ we also prove that the condition is equivalent to $\cos (\theta \mathrm{/}{2}^{e-1})\in F,$ where $\theta$ is the angle between $\mathit{a}$ and $\mathit{b}\mathrm{.}$