Number theory Algebraic number theory Problem

Class number problem

Builds on Quadratic field Class number Binary quadratic form

Problem

For each positive integer $h$ , determine the complete list of imaginary quadratic fields $\Q(\sqrt{-d})$ , with $d$ a positive squarefree integer, whose class number equals $h$ .

Remarks

The problem goes back to Gauss, who stated it in the language of binary quadratic forms. It has three parts: show that the class number tends to infinity as $d \to \infty$ , so that each list is finite; make this effective; and compute the lists.

Status: the class number of $\Q(\sqrt{-d})$ tends to infinity with $d$ (Heilbronn), but this proof gives no effective bound. The case $h = 1$ is the Stark–Heegner theorem, with nine fields, and the case $h = 2$ was solved by Baker and by Stark. Goldfeld, combined with the theorem of Gross and Zagier, gave an effective lower bound for the class number, which reduces each $h$ to a finite computation. Watkins completed the lists for all $h \le 100$ .

The analogue for real quadratic fields is open: it is not known whether infinitely many real quadratic fields have class number $1$ .

Reading path

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Tag	`KTGAS9DGRX`
URI	`m2://KTGAS9DGRX`
Kind	problem
Writing	W0 stub
MSC	11R29, 11R11
Status	This copy of the site has no build data. Run `m2 status KTGAS9DGRX`.
About	Class number, Quadratic field
Uses	Quadratic field, Class number, Binary quadratic form
Changed	2 November 2026

Class number problem

Problem

Remarks

Reading path

Where this leads

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