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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 6 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 The first 60 powers of (3 + 4i)/5. The powers of (3 + 4i)/5 marked on the unit circle, each a further turn by the same angle, with the power that comes closest to returning to 1 marked.

On the circle and never home

Every root of unity lies on the unit circle, and so does the point (3 + 4i)/5 — yet no power of it ever returns to 1. A root of a whole-number polynomial of degree ten does the same. What forces a point home is a condition on the numbers its polynomial ties it to, and how far one of them may stray is a question open since 1933.

algebra · Roots of unity Six thirtieths of a turn that add to nothing. On the left a unit circle with the roots of unity in a vanishing sum marked and coloured by origin; on the right the same unit vectors placed head to tail, returning to their start.

The sums of roots of unity that add to nothing

All n of the n-th roots of unity add to zero, and so does any regular polygon among them, turned. Those are not the only vanishing sums: six thirtieths of a turn close into a loop with no polygon in them. Which counts of roots can close at all is decided by the prime factors of n — no seven fifteenths ever add to zero — and the same question counts where the diagonals of a regular polygon cross.

algebra · Roots of unity The 3 periods of the 13th roots of unity, and the equation they solve. Roots of unity modulo 13 grouped by the cosets of the index-3 subgroup; the periods 0.274, 1.377, −2.651 are the roots of x³ + x² − 4x + 1.

The sums that obey a smaller equation

The twelve non-trivial thirteenth roots of unity satisfy an equation of degree twelve. Split them into three groups of four — the right three groups — and add each group: the three sums are the roots of x³ + x² − 4x + 1, an equation of degree three with whole-number coefficients. Gauss called such sums periods, found one for every divisor of p − 1, and used them to build the seventeen-gon from four quadratic equations.

algebra · Roots of unity The primitive 15th roots of unity, raised to a power and added. c₁₅(1) = 1; c₁₅(3) = −2; c₁₅(5) = −4; c₁₅(6) = −2; c₁₅(10) = −4; c₁₅(15) = 8.

The sums of primitive roots are always whole

Take the roots of unity of order exactly q, raise each to the power n and add them. The answer is always a whole number, it depends on n only through what n shares with q, and as n varies the sums behave like the sines and cosines of a Fourier series — so well that Ramanujan could rebuild the sum of the divisors of any number from them, and prove that their weighted total is nought, a fact that at n = 1 is the prime number theorem.

algebra · Roots of unity

Named alongside it

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

Roots of unityComplex numbersMinimal polynomialPrimitive elementRegular polygonTotientAlgebraic integerConjugateConstructible numberCosetCyclic groupExact arithmetic

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