Rational Quartic Reciprocity
نویسنده
چکیده
In 1985, K. S. Williams, K. Hardy and C. Friesen [11] published a reciprocity formula that comprised all known rational quartic reciprocity laws. Their proof consisted in a long and complicated manipulation of Jacobi symbols and was subsequently simplified (and generalized) by R. Evans [3]. In this note we give a proof of their reciprocity law which is not only considerably shorter but also sheds some light on the raison d’être of rational quartic reciprocity laws. For a survey on rational reciprocity laws, see E. Lehmer [7]. We want to prove the following Theorem. Let m ≡ 1 mod 4 be a prime, and let A, B, C be integers such that A2 = m(B2 + C2), 2 |B, (A,B) = (B,C) = (C,A) = 1, A+B ≡ 1 mod 4.
منابع مشابه
Rational Quartic Reciprocity Ii
for every prime p ≡ 1 mod 4 such that (p/pj) = +1 for all 1 ≤ j ≤ r. This is ’the extension to composite values of m’ that was referred to in [3], to which this paper is an addition. Here I will fill in the details of a proof, on the one hand because I was requested to do so, and on the other hand because this general law can be used to derive general versions of Burde’s and Scholz’s reciprocit...
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