Date: Sun, 18 May 1997 09:11:10 -0600
From: rjc@maths.ex.ac.uk
Subject: Re: Class numbers
Newsgroups: sci.math

In article <863881688.womack@ox.compsoc.org.uk>,
  womack@ox.compsoc.org.uk (Tom Womack) wrote:
>
> What is it that makes the number of quadratic forms with a given
> discriminant x and the size of the quotient group of integer ideals by
> prime ideals in the field Q[sqrt(x)] the same?
>
> (I've just started a course on algebraic number theory; I've come
> across the term 'class number' in both contexts, and I don't see where
> the relation is)

One can pair off ideal classes and classes of forms in a natural manner.
If I is an ideal of O_K, the ring of integers of K = Q(sqrt(x)), then
I = Z alpha + Z beta for some alpha and beta (one should ensure that
alpha and beta are correctly "oriented"). Then for integers x and y,
x alpha + y beta lies in I and its so norm is divisible by the norm of
I. Thus (x,y) |-> N(x alpha + y beta)/NI is an integral quadratic form
whose discriminant equals that of K. As the choice of (oriented) alpha
and beta alters this form by a proper linear substitution only the
class of the form depends on the ideal. Also ideals in the same
(proper) ideal class give the same classes of forms. For details see e.g.,
Primes of the Form x^2 + ny^2 by David Cox (Wiley 1989).

Robin John Chapman                      "For years I believed Pathos was
Department of Mathematics                one of the Three Musketeers,
University of Exeter, EX4 4QE, UK        Fellatio was a character in
rjc@maths.exeter.ac.uk                   Hamlet,..."
http://www.maths.ex.ac.uk/~rjc/rjc.html    Iain Banks - The Wasp Factory

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