Real Quadratic Fields
The contrasting case of real quadratic fields is very different, and much less is known. That is because what enters the analytic formula for the class number is not h, the class number, on its own — but h log ε, where ε is a fundamental unit. This extra factor is hard to control. It may well be the case that class number 1 for real quadratic fields occurs infinitely often.
The Cohen-Lenstra heuristics are a set of more precise conjectures about the structure of class groups of quadratic fields. For real fields they predict that about 75.446% will have class number 1, a result that agrees with computations.
Read more about this topic: Class Number Problem
Famous quotes containing the words real and/or fields:
“Barney McKlennan was a real man. When he kissed you, you stayed kissed.”
—Lamar Trotti (1898–1952)
“During the first World War women in the United States had a chance to try their capacities in wider fields of executive leadership in industry. Must we always wait for war to give us opportunity? And must the pendulum always swing back in the busy world of work and workers during times of peace?”
—Mary Barnett Gilson (1877–?)