Editing Quadratic reciprocity (section)

==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reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.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''".

*{{cite book
 | last = Gauss | first = Carl Friedrich
 | author-link = Carl Friedrich Gauss
 | translator-last1= Clarke | translator-first1 = Arthur A.  
 | title = Disquisitiones Arithemeticae
 | publisher = [[Springer Science+Business Media|Springer]]
 | place = New York
 | year = 1986
 | isbn = 0-387-96254-9| edition = Second, corrected
 }}
*{{cite book
 | last = Gauss | first = Carl Friedrich
 | author-link = Carl Friedrich Gauss
 | translator-last1= Maser | translator-first1 = Hermann 
 | title = Untersuchungen über höhere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory)
 | publisher = Chelsea 
 | place = New York
 | year = 1965
 | isbn = 0-8284-0191-8| edition = Second
 }}

The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1&ndash;23 and the second §§ 24&ndash;76. Footnotes referencing these are of the form "Gauss, BQ, § ''n''". 
*{{citation
 | last1 = Gauss | first1 = Carl Friedrich
 | title = Theoria residuorum biquadraticorum, Commentatio prima
 | publisher = Comment. Soc. regiae sci, Göttingen 6
 | location = Göttingen
 | year = 1828}}
*{{citation
 | last1 = Gauss | first1 = Carl Friedrich
 | title = Theoria residuorum biquadraticorum, Commentatio secunda
 | publisher = Comment. Soc. regiae sci, Göttingen 7
 | location = Göttingen
 | year = 1832}}

These are in Gauss's ''Werke'', Vol II, pp.&nbsp;65&ndash;92 and 93&ndash;148. German translations are in pp.&nbsp;511&ndash;533 and 534&ndash;586 of ''Untersuchungen über höhere Arithmetik.''

Every textbook on [[Number theory#Elementary number theory|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 of quadratic reciprocity|proofs for the quadratic reciprocity law]].

Kenneth Ireland and [[Michael Rosen (mathematician)|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.&nbsp;202) says it all
:<blockquote>Count the number of proofs to the law of quadratic reciprocity given thus far in this book and devise another one.</blockquote>

*{{citation
 | last1 = Bach | first1 = Eric
 | last2 = Shallit | first2 = Jeffrey 
 | title = Algorithmic Number Theory (Vol I: Efficient Algorithms)
 | publisher = [[The MIT Press]]
 | location = Cambridge
 | year = 1966
 | isbn = 0-262-02405-5}}
*{{Citation
 | last = Edwards
 | first = Harold | author-link = Harold Edwards (mathematician)
 | title = Fermat's Last Theorem
 | publisher = [[Springer Science+Business Media|Springer]]
 | location = New York
 | year = 1977
 | isbn = 0-387-90230-9}}
*{{citation
 | last1 = Lemmermeyer | first1 = Franz
 | title = Reciprocity Laws: From Euler to Eisenstein
 | publisher = [[Springer-Verlag]]
 | series = Springer Monographs in Mathematics
 | location = Berlin
 | mr = 1761696
 | year = 2000
 | isbn = 3-540-66957-4
 | doi= 10.1007/978-3-662-12893-0}}
*{{citation
 | last1 = Ireland | first1 = Kenneth
 | last2 = Rosen | first2 = Michael
 | title = A Classical Introduction to Modern Number Theory (second edition)
 | publisher = [[Springer Science+Business Media|Springer]]
 | location = New York
 | year = 1990
 | isbn = 0-387-97329-X}}