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He outlined a proof of the fact that there are only finitely many algebraic number fields with a given discriminant.

I wish to know whether this fact is somehow related to the assertion that, for any C>0, the number of complex quadratic fields with class number less than C is finite...

Thanks in advance for your answers.

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    If Hermite's theorem proved this, then it would also prove that there are finitely many *real* quadratic fields of class number 1, which is expected (but not known) to be false.2011-12-18

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