Titlebook: Reciprocity Laws; From Euler to Eisens Franz Lemmermeyer Book 2000 Springer-Verlag Berlin Heidelberg 2000 Algebra.Elliptic functions.Gaus a

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Reciprocity Laws978-3-662-12893-0Series ISSN 1439-7382 Series E-ISSN 2196-9922

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Cyclotomic Number Fields,This chapter is devoted to some proofs of the quadratic reciprocity law that make use of the arithmetic of cyclotomic number fields.

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Quartic Reciprocity,In Chapter 5 we have already seen a lot about quartic reciprocity and its applications to rational number theory; these rational laws, however, do not suffice to solve every “rational” problem where quartic reciprocity is involved, as the following example shows.

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,Gauss’s Last Entry,It is well known that Gauss recorded many of his discoveries in a diary; it ends with the ‘Last Entry’ from July 9, 1814, which reads as follows. (see [Kle]):

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The Genesis of Quadratic Reciprocity,und very early on (see [Ene]) — in connection with the problem of characterizing perfect squares — the history of modern number theory starts with the editions of the books of Diophantus, in particular with the commented edition by Bachet in 1621.

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