Class Number

The class number of a number field K is the order of its ideal class group, denoted h_K. It measures the failure of unique factorization in the ring of integers O_K: the class number is 1 if and only if O_K is a unique factorization domain. For the rational numbers, the class number is trivially 1. For quadratic fields, the class number is already a source of deep mystery: it grows irregularly with the discriminant, and its exact value is connected to the residue of the Dedekind zeta function at s = 1 through the analytic class number formula.

The class number problem — to determine all imaginary quadratic fields with a given class number — was solved for class number 1 by Heegner, Stark, and Baker, who identified exactly nine such fields. The general problem remains open. For real quadratic fields, the class number is even more mysterious, connected to the regulator and the continued fraction expansion of √d. The class number is not merely a statistical invariant; it is the organizational fingerprint of a number field, encoding how the field's multiplicative structure deviates from the ideal of unique factorization. Fields with the same degree and similar discriminants can have wildly different class numbers, suggesting that class number is a measure of organizational complexity, not of linear extension.