Algebraic Characterizations of Angle Multisections over Rings

TL;DR

This paper generalizes the rational angle bisection problem over rings, characterizing when two vectors form an angle with a sequence of sector vectors via polynomial roots in the field of fractions.

Contribution

It provides algebraic criteria involving polynomial roots and cosines for angle multisection over rings, extending previous rational bisection results.

Findings

Condition equivalent to existence of a polynomial root in the field of fractions.

For R = Z, condition relates to divisors of the polynomial's constant term.

When m is a power of two, condition involves cosine of halved angles in the field.

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 $\boldsymbol{b}.$