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I know that in general the number of ideal classes are not 1, and that there are only 9 imaginary quadratic number fields which are principal ideal domains, i.e. $\mathbb(Q(\sqrt{-m}))$ where m is 1, 2, 3, 7, 11, 19, 43, 67, 163, and that Gauss had conjectured that there are infinitely many real quadratic number fields which are of class number 1. Although this sounds natural, since the order of unit groups in a real quadratic number field is infinity, while that of an imaginary one is finite, I cannot give a proof. So my question is this:

How many real quadratic number fields are principal ideal domains?

If there is any error, such as this is already known, or there is a wild discussion on this topic, please let me know, thanks very much.

In addition:
Even if this is not solved, I would like to get some modern, or recent, research, or paper, to extend my horizon, so it will be best to have a reference which discusses the number of ideal classes of algebraic number fields, best regards here.

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Broadly speaking, the information you gave in your question is up-to-date: ever since Gauss, number theorists have believed there should be infinitely many real quadratic fields of class number one, but we are not any closer to being able to prove this than Gauss was, to the best of my knowledge. Moreover you have identified the key problem: the fact that a real quadratic field has an infinite unit group means that there is an additional quantity in the analytic class number formula, the regulator, which is not present in the imaginary quadratic case, and it is hard to separate out the respective contributions of the class number and regulator.

In recent years a trend of research has been towards making increasingly precise quantitative conjectures about the expected behavior of the class number -- or class group -- of a real quadratic field of discriminant $D$. A lot of this comes under the heading of Cohen-Lenstra heuristics.

Since you asked for some pointers to recent work, here are two interesting papers that I found (from a google search, I fully admit):

I. Stephens and Williams, Computation of real quadratic fields with class number one.

II. Ono, Indivisibility of class numbers of real quadratic fields. (Wayback Machine), doi:10.1023/A:1001533613223

There is certainly plenty more literature available. A MathSciNet search for "class number" AND "real quadratic field" calls up precisely [well, not for long!] 500 papers...

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    $\begingroup$ It should be "the regulator, which is not present in the imaginary quadratic case". $\endgroup$
    – lhf
    Mar 22, 2011 at 13:25
  • $\begingroup$ Thanks @Pete L. Clark, and I have computed for a while the class number for imaginary quadratic number fields up to $\mathbb(Q(\sqrt{-30}))$ by the class number formula of Dirichlet which already makes me feel that there should be an easier way of computing this prominent quantity for a simple case like this. And, in view of the fact that we have the lists for prime numbers, is there a list for class numbers of algebraic number fields, either quadratic or of higher degree? Thanks. $\endgroup$
    – awllower
    Mar 22, 2011 at 13:29
  • $\begingroup$ The paper II seems to be now located here: mathcs.emory.edu/~ono/publications-cv/pdfs/046.pdf But perhaps adding a link to Internet Archive is an easier way to prevent link rot than changing the link each time the author changes their homepage. $\endgroup$
    – Martin Sleziak
    Dec 23, 2016 at 16:45

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