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2015 | OriginalPaper | Buchkapitel

12. Proofs Using Results from Cyclotomy

verfasst von : Oswald Baumgart

Erschienen in: The Quadratic Reciprocity Law

Verlag: Springer International Publishing

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Abstract

In Chap. 5 we have collected the proofs that are based on theorems from cyclotomy. This theory was founded by Gauss when he was looking for another proof of his fundamental theorem. Already in 1796 [24] he announced the construction of the 17-gon. Apart from the fundamental theorems on imaginary numbers and functions, Gauss derived three (or, if you want, four) different proofs of the reciprocity law.

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Vorheriges Kapitel Eisenstein’s Proofs Using Complex Analysis

Nächstes Kapitel Proofs Based on the Theory of Quadratic Forms

This deficiency (i.e., the lack of the sign) has cast a cloud over everything else I have found, and since four years hardly a week has passed in which I did not make a negative attempt of solving this knot. But all the brooding, all the searching was in vain, and each time I was forced to put down the pen in sorrow. Finally, a few days ago, I was successful – but not due to my arduous search but only by the grace of God, as I would say. Like lightning strikes the riddle was solved; I myself would be unable to tell you the connection between what I knew before, in my last attempts – and the idea by which I succeeded. Curiously the solution of the problem now appears to be easier than many other results which have not cost me as many days as this problem cost me years, and certainly no one will get any idea about the tight squeeze I was in for so long when I eventually present this matter.

The difficulty of fully understanding the exact reason for the success of these delicate considerations which the illustrious author uses for these ingenious transformations made me investigate whether one could not resolve the same question without using them, and I have succeeded …

[FL] Here Baumgart quotes a sentence from Heine’s article, which does not seem to be relevant to our topic.

Using the same principles, Lebesgue also proved various formulas from Jacobi’s Elliptische Functionen, p. 186.

[BEW]

B. Berndt, R. Evans, K.S. Williams, Gauss and Jacobi sums, John Wiley & Sons, 1998 123

[Cass]

B. Casselman, Dirichlet’s evaluation of Gauss sums Enseign. Math. (2) 57 (2011), 281–301 123

[Cast]

W. Castryck, A shortened classical proof of the quadratic reciprocity law, Amer. Math. Monthly 115 (2008), 550–551 123

[Gen]

A. Genocchi, Note sur la théorie des residus quadratiques, Mém. cour. et mém. des savants étrangers Acad. Roy Sci. Lettres Belgique 25 (1851/53), 54 pp 123

[IR]

K. Ireland, M. Rosen, A classical introduction to modern number theory, Springer Verlag, 2nd. ed. 1990 123

[Ler]

M. Lerch, Zur Theorie der Gaußschen Summen, Math. Ann. 57 (1903), 554–567 123

[Mur]

M.R. Murty, Quadratic reciprocity via linear algebra, Bona Mathematica 12, No. 4 (2001), 75–80 123

[Sal]

M. Salvadori, Esposizioni delle teoria delle somme di Gauss et di alcuni teoremi di Eisenstein, Diss. Freiburg (CH) 1904, 116 pp 123

[Sch1]

M. Schaar, Mémoire sur la théorie des residus quadratiques, Acad. Roy. Sci. Lettres Beaux Arts Belgique 24 (1852), 14 pp; cf. p. 123

[Sch2]

M. Schaar, Recherches sur la théorie des residus quadratiques, Acad. Roy. Sci. Lettres Beaux Arts Belgique 25 (1854), 20 pp; cf. p. 123

[Sch]

I. Schur, Über die Gauß’schen Summen, Göttinger Nachr. 1921, 147–153 123

[Sha]

D. Shanks, Two theorems of Gauss, Pac. J. Math. 8 (1958), 609–612 123

[Wat]

W. Waterhouse, The sign of the Gaussian sum, J. Number Theory 2 (1970), 363 123

Titel: Proofs Using Results from Cyclotomy
verfasst von: Oswald Baumgart
Verlag: Springer International Publishing
Buch: The Quadratic Reciprocity Law
Print ISBN: 978-3-319-16282-9

Electronic ISBN: 978-3-319-16283-6

Copyright-Jahr: 2015
DOI: https://doi.org/10.1007/978-3-319-16283-6_12

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