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Concept

Class number

The number of classes of quadratic forms, or of ideals, sharing one discriminant, counted by reducing each form to a smallest representative. It is one exactly when factorisation into primes is unique, and it grows roughly like the square root of the discriminant.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

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 How often a prime divides the class number. p 3: 39.1% against 1/p 33.3% and predicted 44.0%; p 5: 22.6% against 1/p 20.0% and predicted 24.0%; p 7: 15.3% against 1/p 14.3% and predicted 16.3%; p 11: 9.1% against 1/p 9.1% and predicted 9.9%.

Class groups drawn at random

The class number of an imaginary quadratic field measures how badly unique factorisation fails there, and it looks arbitrary: 1, 2, 4, 3, 5, 12, 1. Count how often 3 divides it and the answer is not one time in three but noticeably more — heading, by a conjecture of Cohen and Lenstra, for 44 per cent. The reason is that class groups behave like finite groups chosen at random with each group weighted by one over its number of symmetries, so groups with few symmetries, like the cyclic group of order 3, turn up more often than the naive count suggests.

number · Unique factorisation A ball counted slice by slice. Ball of radius 7.5: slices 0:177, 1:177, 2:169, 3:145, 4:129, 5:97, 6:69, 7:21; total 1791, volume 1767.146.

The dots a ball catches

Count the lattice points inside a ball and compare with its volume. In five dimensions the error is about the size of one spherical shell, and that is a theorem; in four a logarithm sneaks in; in three and two the true size of the error is unknown. The difficulty runs backwards because the arithmetic of sums of squares is most erratic when there are fewest squares to add.

discrete · Pick theorem

Named alongside it

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

Class groupQuadratic formDiscriminantAsymptoticsAutomorphismDimensionDivisor functionError termGenusGenus theoryIdealIdoneal number

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