Abstract

A simple proof of Yamamoto's reciprocity law is given. Let p and q be distinct odd primes with p _ q _ 1 (mod 4). Define the symbol [p, q] =i by (q-1)/4 (pq)12 (mod q), where h(pq) is the classnumber of the real quadratic field Q(V/R) and e(pq) = 2 (t + u,/jj) > 1 is its fundamental unit (t, u positive integers). Yamamoto's reciprocity law [5, Theorem 3] states that [p, q] = [q, p]. We give a simple proof of a slightly stronger result: Theorem. Let p and q be distinct primes with p _ q _ 1 (mod 4); let A(p, q) be the number of pairs (x, y) of integers satisfying 0<x<, ?<Y< 1, qx <py, 5 p)_( where (x/p) is the familiar Legendre symbol. Then [p5 q] = [q p] = A(p,q)(P) q The following lemma is well known; notation is as above. Lemma. h(pq) is even; furthermore (a) h(pq) 0 (mod 4), if (q) = 1, (q)4 = ()4 [1, Theorem 4; 4, p. 603], (b) (t)2 pq( u)2 = +1, if (p) = 1, (q)4 = -(p)4 [4, p. 603], (c) h(pq) 2 (mod 4), (t)2 pq(u)2 =-1, if (-) =-1 [1, Theorem 4; 3, ?3]. Received by the editors June 7, 1989 and, in revised form, March 26, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 1 lA1 5; Secondary 1 IRI 1.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.