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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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    @Fredrik: You gave suitable background and context so that the answer wasn't an open ended "Tell me everything you can think of about real quadratic/imaginary quadratic number fields." In my mind, specifying exactly what led you to your question makes a big difference, because then the answers could be direct and relevant.2011-03-22

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  • 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.