Set complexity of construction of a regular polygon

TL;DR

The paper investigates the complexity of constructing regular polygons using algebraic operations, proving that all p-th roots of unity can be generated from 1 with a bounded number of steps for Fermat primes.

Contribution

It establishes a specific upper bound on the number of algebraic operations needed to construct p-th roots of unity from a single element, for Fermat primes.

Findings

All p-th roots of unity can be obtained from 1 using 12 p^2 operations for Fermat primes.

The result differs from standard algorithmic complexity estimates for computing roots of unity.

Provides a new perspective on the algebraic complexity of geometric constructions.

Abstract

Given a subset of $\mathbb C$ containing $x,y$, one can add $x + y,\,x - y,\,xy$ or (when $y\ne0$) $x/y$ or any $z$ such that $z^2=x$. Let $p$ be a prime Fermat number. We prove that it is possible to obtain from $\{1\}$ a set containing all the $p$-th roots of 1 by $12 p^2$ above operations. This result is different from the standard estimation of complexity of an algorithm computing the $p$-th roots of 1.