Concept

Algebraic integer

A number that is a root of a polynomial with whole coefficients and leading coefficient 1. Such numbers are closed under sums and products, and in a field like the one generated by √5 they form the ring that number theory uses in place of the integers.

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

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

The integers of ℚ(√5), with ℤ[√5] inside them. Points a + bφ plotted against their conjugates for small whole a and b, with the index-two sublattice ℤ[√5] filled and the basic cells of both lattices shaded, of areas √5 and 2√5.

The integers a field contains

Inside the field of numbers a + b√5, the obvious integers are those with whole a and b. They are not all of them: the golden ratio has a one-half in it and satisfies x² = x + 1, a monic equation with whole coefficients, exactly as an integer should. The right integers form a lattice twice as dense as the obvious one — and a whole-number matrix proves they are closed under addition.

algebra · Field extensions

Named alongside it

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

Minimal polynomial Characteristic polynomial Conjugate Cyclotomic polynomial Determinant Field extension Gaussian integers Golden ratio Irrational rotation Lattice Mahler measure Pigeonhole principle

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