Concept

Primitive element

A member of a group whose powers run through the whole group, or of a field extension that generates all of it. Its existence turns a structure into a list of powers, so that multiplication becomes addition of exponents.

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

The non-zero elements of GF(16) as the powers of one of them. A ring of the field's non-zero elements in the order the powers of a primitive element produce them, beside a table of exponents.

Every element is a power of one of them

Pick the right element of a finite field and its powers run through every other non-zero element exactly once before returning to one. Multiplication becomes addition of exponents, and a table of q − 1 entries replaces the whole multiplication table.

computation · finite fields

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.

Cyclic group Totient Complex numbers Counting argument Cyclotomic polynomial Discrete logarithm Factorisation Fermats little theorem Finite field Modular arithmetic Regular polygon Roots of unity

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