class number

If the ideal class group is a precise diagnosis of how unique factorization fails, the class number is the single most important headline statistic: how many distinct ideal classes are there? When it equals 1, the ring of integers factors just as nicely as the ordinary integers; when it is large, the arithmetic is correspondingly wilder.

The class number h_K of a number field K is the order of its ideal class group Cl(O_K). It is always a finite positive integer, a theorem made effective by the Minkowski bound, which guarantees every ideal class contains a representative of small norm and so reduces the computation to a finite search. The class number equals 1 exactly when O_K is a principal ideal domain.

Class numbers are mysterious and deeply studied. Gauss conjectured there are only finitely many imaginary quadratic fields of each class number, finally proved for h = 1 (there are exactly nine) and in general by Goldfeld, Gross and Zagier. The analytic class number formula expresses h_K through the residue of the Dedekind zeta function at s = 1, tying class numbers to L-functions.