an:02751761 ERAM 002.0047cj Jacobi, C. G. J. A few remarks on cubic residues LA J. Reine Angew. Math. 2, 66-69 (1827). 00220852 1827
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11A15 cubic residues; cubic reciprocity law Jacobi remarks that Gau?? has announced a memoir on biquadratic residues, in which he will prove a criterion for $$2$$ to be a biquadratic residue of primes $$p$$. He observes that for cubic residuacity of primes $$p = 3n+1$$ one has to consider the representations $$4p = L^2 + 27M^2$$. He announces the following theorem: If $$p$$ and $$q$$ are prime numbers of the form $$3n+1$$, with $$4p = L^2 + 27M^2$$, and if $$x$$ is an integer with $$x^2 + 3 \equiv 0 \pmod q$$, then $$q$$ will be a cubic residue with respect to $$p$$ if and only if $$\frac{L+3mx}{L-3Mx}$$ is a cubic residue with respect to $$p$$. His second theorem deals with primes $$q = 6n-1$$; he calls integers $$x$$ with $$x^{\frac{q+1}3} \equiv 1 \pmod p$$ cubic residues of $$q$$, and announces the following result: If $$p$$ is a prime of the form $$6n+1$$, if $$4p = L^2 + 27M^2$$, and if $$q$$ is a prime of the form $$6n-1$$, then $$q$$ will be a cubic residue with respect to $$p$$ if and only if $$\frac{L+3m\sqrt{-3}}{L-3M\sqrt{-3}}$$ is a cubic residue with respect to $$p$$. In addition he remarks that if $$p = 3n+1$$ satisfies $$4p = L^2 + 27M^2$$, then $$L$$ is the minimal remainder modulo $$p$$ of $$- \frac{(n+1)(n+2)\cdots 2n}{1 \cdot 2 \cdots n} = - \binom{2n}{n}$$ that has the form $$3k+1$$, and gives a similar result for primes $$p = 7n+1$$. Jacobi presented the proofs of these results in his lectures; see [\textit{F. Lemmermeyer} (ed.) and \textit{H. Pieper} (ed.), Vorlesungen ??ber Zahlentheorie. Carl Gustav Jacob Jacobi, Wintersemester 1836/37, K??nigs\-berg. Augsburg: ERV Dr. Erwin Rauner Verlag (2007; Zbl 1148.11003)]. Franz Lemmermeyer (Jagstzell) (2015) Zbl 1148.11003