95m:11010 11A15

Lemmermeyer, Franz Rational quartic reciprocity. Acta Arith. 67 (1994), no. 4, 387–390. In 1985, K. S. Williams, K. Hardy and C. Friesen published a reciprocity formula that comprised all known quartic reciprocity laws. Their proof was simplified and generalized by R. Evans. In this paper the author gives a proof of their reciprocity law which is considerably shorter and also sheds some light on the raison d’etre of rational quartic reciprocity laws. The author proves that if m ≡ 1 mod 4 is a prime and A, B, C are pairwise relatively prime integers such that A^{2} = m(B^{2} + C^{2}), 2 | B, and A + B ≡ 1 mod 4, then for every odd prime p > 0 such that (\frac{m}{p}) = +1,

(\frac{A + B\sqrt{m}}{p}) = (\frac{p}{m})_{4}.

Safwan Akbik (1-HOFS)

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