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Bumped in Stillwell's book "Elements of Number Theory" into "The real quadratic fields with unique prime factorization are still not known ...". But doesn't $\mathbb{Q}[\sqrt{2}]$'s ring of integers $\mathbb{Z}[\sqrt{2}]$ have that property?

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    $\begingroup$ Yes, it has unique factorization. $\endgroup$ – André Nicolas Jul 17 '15 at 23:31
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    $\begingroup$ I suspect that what was written was that nobody knows if there are infinitely many real quadratic fields with this property. As you say, there are certainly known examples. $\endgroup$ – lulu Jul 17 '15 at 23:31
  • $\begingroup$ Here's a literal quote from the book: "The real quadratic fields with unique prime factorization are still not known, nor is it known whether there are infinitely many of them". $\endgroup$ – user75619 Jul 17 '15 at 23:38
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    $\begingroup$ Would it help if we wrote "Not all ____ are known. In particular we do not know whether there are infinitely many" Let me add that many examples are known, and it seems likely that there are infinitely many. $\endgroup$ – Will Jagy Jul 17 '15 at 23:44
  • $\begingroup$ I can now see how "are still not known" when put in the context can mean "not understood as well as imaginary quadratic fields". But as someone who's just getting familiarized oneself with these concepts, the wording could have been better. Anyway, thanks everyone for clarification. $\endgroup$ – user75619 Jul 17 '15 at 23:51
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see https://oeis.org/A003172 and https://en.wikipedia.org/wiki/List_of_number_fields_with_class_number_one#Real_quadratic_fields
for some detail.

The difference, quite enormous, from imaginary quadratic fields is that those have only a handful of class number one, and it was a huge endeavor to prove the known list complete. https://en.wikipedia.org/wiki/Stark%E2%80%93Heegner_theorem

For real, everyone expects that there are infinitely many. However, in a few cases, notably $$ 229,\; 257, \; 401, $$ despite being primes congruent to $1 \pmod 4,$ the class number is larger than one (although still odd). If the thing gave class number one for all such primes, we would be able to say that we knew there were infinitely many, but it does not always work.

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  • $\begingroup$ Your final reference gives me a 404 error, pls fix. Thank you. $\endgroup$ – Oscar Lanzi Jul 22 '17 at 13:30
  • $\begingroup$ @OscarLanzi they discontinued the blog and did not give any alternate place for the posts. $\endgroup$ – Will Jagy Jul 22 '17 at 16:22
  • $\begingroup$ Can you or I delete the lost reference then? $\endgroup$ – Oscar Lanzi Jul 22 '17 at 16:24
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    $\begingroup$ Yes, go ahead... $\endgroup$ – Will Jagy Jul 22 '17 at 16:26
  • $\begingroup$ Alas, the sad story plays out. The moral that you likely know: always give important summaries with your references so that at least some information survives. $\endgroup$ – Oscar Lanzi Jul 22 '17 at 16:33

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