Algebraic Characterizations of Angle Multisections over Rings

Abstract

Let $n,$ $m \geq 2$ be integers, and let $R$ be a subring of $\mathbb R$ with field of fractions $F.$ In this article, we generalize the rational angle bisection problem previously proposed by the author to the following problem: which linearly independent vectors $\boldsymbol{a},$ $\boldsymbol{b} \in R^n$ form an angle with a sequence of $m$-sector vectors lying in $R^n$? When $\boldsymbol{a}$ and $\boldsymbol{b}$ 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 $R.$ 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 $e \geq 1,$ we also prove that the condition is equivalent to $\cos (\theta /2^{e-1}) \in F,$ where $\theta$ is the angle between $\boldsymbol{a}$ and $\boldsymbol{b}.$