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Sorry for the vague title... I've proved a number theoretical result for the imaginary quadratic fields (it was already known for the rationals). I think it would be much easier to sell if I could prove it for all quadratic fields. I think to do it will require using some special properties of the field, as proving it for general number fields seems very difficult...

What special properties do the real quadratic fields have, that are not enjoyed by a general number field?

Any and all suggestions welcome. Also, obviously ones which apply to all quadratic fields would also be useful. My result for the imaginary quadratic fields relies on the fact that they have only finitely many units, which means that the norm is 'well behaved' in some sense, but this does not help me for real ones.

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This is a ridiculously vague question. –  Arturo Magidin Mar 22 '11 at 13:51
    
Hmmm I was afraid that might be the case. I'm not really sure how to make it less vague without going into a lot of detail about my problem though. Thought it was worth a shot. –  samian Mar 22 '11 at 14:25
    
well add detail about your problem if you want anyone to say anything... –  yoyo Mar 22 '11 at 14:52
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@samian: It looks as follows to me: You have a result; you don't want to say what it is (perhaps for fear of being scooped; fair enough, perhaps); at the same time, you're not sure it is sufficiently important, so you want to extend it. So you are asking people to list say everything they can think of about the fields you want to extend it to, perhaps in the hope that you will see something that will either help you extend it, or that will show what you want is already known. And you want them to do it blidnly, without any idea where you want to go. It's not only vague, it's rather unfair. –  Arturo Magidin Mar 22 '11 at 14:58
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@samian: If your vagueness is indeed because you don't want to disclose your problem, then my suggestion is: write it up, put it up in a preprint server (to establish your priority), submit it if you want, and then you'll be free to discuss it with your priority well-established. If you didn't discuss it because it would take too long, then perhaps this is simply not a suitable question to ask in a site like this, but something to discuss face-to-face during conferences. They still have those, I believe. –  Arturo Magidin Mar 22 '11 at 15:18
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1 Answer

  • The complex quadratic fields have finitely many units, but the real quadratic fields have infinitely many.

  • There are only a handful of complex quadratic UFDs but nobody knows yet how many real quadratic UFDs there are.

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