From: bernardi@math.jussieu.fr (Dominique Bernardi) Newsgroups: sci.math.research Subject: origin of a proof of the quadratic reciprocity law Date: Wed, 09 Dec 1998 08:57:05 +0100 I would like to have references for this proof that I read some time ago in a newsgroup (sci.math or sci.math.research) For p and q two different odd primes, the following properties are equivalent - p is a square mod q - x |-> px is an even permutation of Z/q/Z - z |-> z^p is an even permutation of the q-th roots of unity in an algebraic closure of Fp - The galois group of X^q - 1 over F_p is contained in the alternating group A_q - The discriminant of the polynomial X^q-1 is a square in F_p - (-1)^((q-1)/2)q is a square mod p -- Dominique Bernardi, Théorie des Nombres Institut de Mathématiques - Université Pierre et Marie Curie 4 place Jussieu - F75005 Paris Tel (33-1) 44275441 bernardi@math.jussieu.fr ============================================================================== From: shallit@graceland.uwaterloo.ca (Jeffrey Shallit) Newsgroups: sci.math.research Subject: Re: origin of a proof of the quadratic reciprocity law Date: Thu, 10 Dec 1998 12:32:07 GMT In article , Dominique Bernardi wrote: >I would like to have references for this proof that I read some time ago >in a newsgroup (sci.math or sci.math.research) > >For p and q two different odd primes, the following properties are equivalent > >- p is a square mod q > >- x |-> px is an even permutation of Z/q/Z > >- z |-> z^p is an even permutation of the q-th roots of unity in an >algebraic closure of Fp > >- The galois group of X^q - 1 over F_p is contained in the alternating group A_q > >- The discriminant of the polynomial X^q-1 is a square in F_p > >- (-1)^((q-1)/2)q is a square mod p > >-- >Dominique Bernardi, Thiorie des Nombres >Institut de Mathimatiques - Universiti Pierre et Marie Curie >4 place Jussieu - F75005 Paris Tel (33-1) 44275441 >bernardi@math.jussieu.fr This is essentially exercise 5.45 in my book with Eric Bach, _Algorithmic Number Theory_, MIT Press, 1996. I learned of this method from Jerry Tunnell when I was an undergraduate. Its origin is obscure, but it seems to be a combination of a proof due to Zolotarev (Nouvelles Annales de Mathematiques 11 (1872), 354-362) and Swan (Pacific J. Math. 12 (1962), 1099-1106). Jeffrey Shallit, Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1 Canada shallit@graceland.uwaterloo.ca URL = http://math.uwaterloo.ca/~shallit/ ============================================================================== From: victor@idaccr.org (Victor S. Miller) Newsgroups: sci.math.research Subject: Re: origin of a proof of the quadratic reciprocity law Date: 10 Dec 1998 10:42:10 -0500 See also the paper below, which follows that point of view Rousseau, G. Exterior algebras and the quadratic reciprocity law. Enseign. Math. (2) 36 (1990), no. 3-4, 303--308. -- Victor S. Miller | " ... Meanwhile, those of us who can compute can hardly victor@idaccr.org | be expected to keep writing papers saying 'I can do the CCR, Princeton, NJ | following useless calculation in 2 seconds', and indeed 08540 USA | what editor would publish them?" -- Oliver Atkin ============================================================================== From: "Robert Harrison" Newsgroups: sci.math Subject: Re: "what is a reciprocity law?" Date: Sat, 26 Dec 1998 19:05:49 -0000 The law of quadratic reciprocity, proved by Gauss, is fundamental to number theory. If the congruence equation (== here is the triple bar 'is congruent to' symbol) x^2 == a (mod p) is soluble (i.e. a has a square root in this field) then a is a quadratic residue mod p, otherwise it is a quadratic nonresidue. We write (a/p) = 1 in the first case (a/p) = -1 in the second and if p divides a we write (a/p) = 0. This (-/p) is called the Legendre symbol. The law of quadratic reciprocity states, for p and q distinct odd primes (p/q) = (q/p) unless p == q == 3 (mod 4) in which case (p/q) = -(q/p), or, put another way, (p/q)(q/p) = (-1)^((p-1)(q-1)/4). LQR has many, many proofs, using results from supposedly disparate parts of mathematics such as combinatorics, trigonometry, algebra and fluid dynamics. Any elementary number theory text should explain/prove LQR. james dolan wrote in message <7634gq$p64$1@nnrp1.dejanews.com>... >i would like to know a full reference for an article titled something >like "what is a reciprocity law?" that i believe i read some years ago >in some scholarly or semi-scholarly journal or magazine. > >it was about "reciprocity laws" in number theory. it might have been >in the american mathematical monthly. > > > >-----------== Posted via Deja News, The Discussion Network ==---------- >http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own ============================================================================== From: "Robert Harrison" Newsgroups: sci.math Subject: Re: "what is a reciprocity law?" Date: Mon, 28 Dec 1998 10:42:09 -0000 Tal Kubo wrote in message <766gpq$2le$1@cauchy.math.brown.edu>... >Robert Harrison wrote: >>The law of quadratic reciprocity, proved by Gauss, is fundamental to >>number theory. [...] > >I'm sure the questioner knows this. My guess is that he's interested >in understanding where the n-categorical "higher reciprocity laws" get >their name. > > >>LQR has many, many proofs, using results from supposedly disparate >>parts of mathematics such as combinatorics, trigonometry, algebra >>and fluid dynamics. > ^^^^^^^^^^^^^^ > >Do you mean Stokes' theorem? There is a proof of Weil reciprocity >using Stokes' theorem. It would be interesting if ordinary quadratic >reciprocity could be proved in a similar way. I realised after pressing the post button I was waffling unnecessarily. It was late I was tired and bored. I don't know the fluids proof. Its by Lewy, H. (1946). Waves on Beaches, Bull. Am. Math. Soc. 14 207-60. Robert