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:
“The real weakness of England lies, not in incomplete armaments or unfortified coasts, not in the poverty that creeps through sunless lanes, or the drunkenness that brawls in loathsome courts, but simply in the fact that her ideals are emotional and not intellectual.”
—Oscar Wilde (1854–1900)
“... my one aim and concentrated purpose shall be and is to show that women can learn, can reason, can compete with men in the grand fields of literature and science ... that a woman can be a woman and a true one without having all her time engrossed by dress and society.”
—M. Carey Thomas (1857–1935)