Cyclotomics, An Introduction

Introduction

The word cyclotomic comes from the Greek. Its literal meaning is, "cut the circle into pieces", and that's what a cyclotomic extension does.

Draw the unit circle in the complex plane, then let p be the n^th root of 1. Use Demoivre's formula to locate p in the complex plane.

In the reals, the n^th root of 1 is 1, so p must be 1 unit away from the origin. In other words, p lies on the unit circle.

The angle of p (relative to the x axis) is multiplied by n, and the result is 0 degrees, since 1 lies on the positive x axis. Therefore the angle of p is 360° divided by n. Actually, any multiple of this angle will do. There are n distinct angles, n points on the unit circle, and n roots of 1.

The n roots of 1 cut the unit circle into equal arcs and define a regular n-gon. When n = 6, for instance, the 6 sixth roots of 1 lie on the unit circle at 60° intervals, and define an inscribed hexagon.