Concept

Cyclotomic polynomial

The polynomial whose roots are the n-th roots of unity of exact order n. Its coefficients are whole numbers, it is irreducible over the rationals, and its degree counts the residues coprime to n — which is what decides whether a regular polygon can be constructed.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The 7 7th roots of unity. 7 points spaced evenly around the unit circle, at the vertices of a regular 7-sided polygon.

The polygon an equation forces

The n solutions of z to the n equals one are the corners of a regular polygon, and nearly everything about them — that they form a group, that they sum to zero, that the equation factors into pieces with whole-number coefficients — is that picture read carefully.

algebra · Roots of unity

The Gauss sum for 13, added one root at a time. Partial sums of the p-th roots of unity signed by the Legendre symbol, drawn as a walk closing on a point at distance root p from the origin.

One sum, squared two ways

Add the p-th roots of unity, each taken with a plus or a minus according to whether its index is a square. The walk that results closes on a point at distance √p from the origin — and squaring that one number, evaluated two different ways, is the reciprocity law.

number · Quadratic reciprocity

Named alongside it

The objects these essays reach for when they reach for this one.

Complex numbers Primitive element Roots of unity Cyclic group Factorisation Gauss sum Legendre symbol Quadratic reciprocity Quadratic residue Regular polygon Totient

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