Date: Wed, 7 Oct 1998 13:49:17 +0100 (BST)
From: "Dr R.J Chapman" <rjc@maths.ex.ac.uk>
To: rusin@math.niu.edu
Subject: Re: Looking for proofs of the local-to-global principle

> I'm really intrigued by the idea of a proof using Dirichlet's theorem.

I'm afraid I know of no proof for Hasse-Minkowski for fields other than
Q not relying on class field theory. I'm sure Serre appeals to Dirchlet
in the proof for Q in his Course of Arithmetic.

Robin
==============================================================================

From: Robin Chapman <rjc@maths.ex.ac.uk>
Newsgroups: sci.math
Subject: Re: Looking for proofs of the local-to-global principle
Date: Fri, 02 Oct 1998 07:34:56 GMT

In article <6v0t2g$f6g$1@gannett.math.niu.edu>,
  rusin@vesuvius.math.niu.edu (Dave Rusin) wrote:
> I am looking for a proof(s) of the local-to-global principle over
> number fields. Specifically, I am interested in proofs of this theorem:
>
> 	For any nonzero integers  a, b, c,  the equation
> 	(*)	a x^2 + b y^2 + c z^2 = 0
> 	has a nonzero solution  (x,y,z)  in integers iff it has
> 	solutions in all completions of the field of fractions.
>
> When "integers" and "field of fractions" have their usual meanings, this
> result was apparently first proved by Legendre and is discussed at length
> by Gauss in Disquisitiones Arithmeticae. There are several styles of
> proof available, some quite simple and appearing in elementary
> number-theory textbooks. Some of these proofs can even be turned into
> effective algorithms.
>
> The theorem is also true when "integers" means "elements in the ring
> of integers in a number field" and "field of fractions" is the number field
> itself. Proofs of this more general statement are harder to track down.
> In Lam's "Algebraic Theory of Quadratic Forms" is a proof which rests
> upon results not proven therein (e.g. the Brauer-Hasse-Noether theorem
> reducing the Brauer group of a field to its local components).
> Reference is made in passing to proofs using e.g. Dirichlet's theorem on
> primes in arithmetic progressions, but I have not seen such a proof.
>
> I am hoping to be able to compute solutions to  (*)  effectively, and
> so would like to find proofs of the theorem which are as concrete as
> possible and which do not rely e.g. on the presence of a Euclidean algorithm
> or on unique factorization. Any pointers to such proofs (or algorithms)
> would be appreciated.
>
For the B-H-N theorem and its corollary the Hasse norm theorem see section
9.6 of Tate's article on global class field theory in Cassels & Frohlich,
Algbraic Number Theory. The book's exercise section outlines the proof
of the local-global principle for n-ary quadratic forms over number fields.

Robin Chapman                           + "They did not have proper
Room 811, Laver Building                -  palms at home in Exeter."
University of Exeter, EX4 4QE, UK       +
rjc@maths.exeter.ac.uk                  -    Peter Carey,
http://www.maths.ex.ac.uk/~rjc/rjc.html +      Oscar and Lucinda

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