Concept

Factorisation

A way of writing an object as a product of smaller objects of the same kind. Whether the decomposition is unique is a property of the surrounding system rather than of the object, and uniqueness fails in arithmetics only slightly larger than the whole numbers.

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

3 roots, and the two numbers the coefficients already knew. The roots of a degree-3 polynomial, found numerically, with the point they average to. That average, and their product, are readable straight off the coefficients without finding the roots at all.

What the coefficients already know

Finding the roots of a polynomial is hard and often impossible in closed form. Reading off their sum, their product and how many of them are real is none of those things — those numbers are sitting in the coefficients, and no root-finding is required to get at them.

algebra · polynomial roots

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

Named alongside it

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

Complex numbers Coefficient Cyclic group Cyclotomic polynomial Degree Discriminant Polynomial Primitive element Regular polygon Root finding Roots Roots of unity

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