23 unsolved problems which David Hilbert proposed to the Congress of Mathematicians in 1900 asked for the "Proof of the most general reciprocity law [f]or an arbitrary number field". Building upon work by Philipp Furtwängler, Teiji Takagi, Helmut Hasse and others, Emil Artin discovered Artin reciprocity in 1923, a general theorem for which all known reciprocity laws are special cases, and proved it in 1927. See also Dedekind zeta function Rational reciprocity law Zolotarev's lemma Notes References The Disquisitiones Arithmeticae has been translated (from Latin) into English and German. The German edition includes all of Gauss's papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. Footnotes referencing the Disquisitiones Arithmeticae are of the form "Gauss, DA, Art. n". The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form "Gauss, BQ, § n". These are in Gauss's Werke, Vol II, pp. 65–92 and 93–148. German translations are in pp. 511–533 and 534–586 of Untersuchungen über höhere Arithmetik. Every textbook on elementary number theory (and quite a few on algebraic number theory) has a proof of quadratic reciprocity. Two are especially noteworthy: Franz Lemmermeyer's Reciprocity Laws: From Euler to Eisenstein has many proofs (some in exercises) of both quadratic and higher-power reciprocity laws and a discussion of their history. Its immense bibliography includes literature citations for 196 different published proofs for the quadratic reciprocity law. Kenneth Ireland and Michael Rosen's A Classical Introduction to Modern Number Theory also has many proofs of quadratic reciprocity (and many exercises), and covers the cubic and biquadratic cases as well. Exercise 13.26 (p. 202) says it all Count the number of proofs to the law of quadratic reciprocity given thus far in this book and devise another one. External links Quadratic Reciprocity Theorem from MathWorld A play comparing two proofs of the quadratic reciprocity law A proof of this theorem at PlanetMath A different proof at MathPages F. Lemmermeyer's chronology and bibliography of proofs of the Quadratic Reciprocity Law (332 proofs) Algebraic number theory Modular arithmetic Number theory Quadratic residue Theorems in number theory