Date: Wed, 7 Oct 1998 13:49:17 +0100 (BST) From: "Dr R.J Chapman" 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 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 -----------== Posted via Deja News, The Discussion Network ==---------- http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own