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Concept

Class group

The group formed by the classes of quadratic forms, or of ideals, of one discriminant under composition. Its size is the class number; its structure says how factorisation fails, and a group with every element of order two marks Euler's convenient numbers.

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

Also named here as class number — the same set of essays touches all of them, so they are one junction rather than several.

The 4 reduced forms of discriminant −84, each in its place. The reduced binary quadratic forms of discriminant −84 (x² + 21y²; 2x² + 2xy + 11y²; 3x² + 7y²; 5x² + 4xy + 5y²) plotted at their roots in the upper half-plane, all inside the modular fundamental region.

Counting the classes that break factorisation

The class number measures how badly unique factorisation fails in a field, and defined through ideals it looks impossible to compute. Gauss computed it by hand, for every field he wanted, by counting quadratic forms — and every form can be squeezed, by changes of variable that keep its values, into exactly one small standard shape.

number · Unique factorisation Class numbers of x² + ny², with Euler's idoneal numbers marked. A scatter of class number against n up to 2000, on a logarithmic scale, with the 65 idoneal numbers marked on the power-of-two levels.

Euler's sixty-five convenient numbers

For some n, whether a prime can be written as x² + ny² is settled by its remainder on division by 4n alone, the way a prime's remainder on division by four settles whether it is a sum of two squares. Euler found sixty-five such n, from 1 to 1,848, called them convenient, and used the largest to prove that 18,518,809 is prime. Every one of them is a number whose class group has no element of order more than two — and whether the list is complete is still not known.

number · Unique factorisation

Named alongside it

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

Class numberDiscriminantQuadratic formGenusIdealIdoneal numberModular groupPrimality testQuadratic fieldRemainderUnique factorisation

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