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и, конечно, на книгу Леммермейера «Reciprocity Laws: From Euler to Eisenstein» полезно обратить внимание
«The history of reciprocity laws is a history of algebraic number theory. This is a book on reciprocity laws, and our introductory remark is placed at the beginning as a warning: in fact a reader who is acquainted with little more than a course in elementary number theory may be surprised to learn that quadratic reciprocity does — in a sense that we will explain — belong to the realm of algebraic number theory. Heeke has formulated this as follows:
Modern number theory dates from the discovery of the reciprocity law. (…) The development of algebraic number theory has now actually shown that the content of the quadratic reciprocity law only becomes understandable if one passes to general algebraic numbers and that a proof appropriate to the nature of the problem can be best carried out with these higher methods.
Naturally, along with these higher methods came generalizations of the reciprocity law itself. It is no exaggeration to say that this generalization changed our way of looking at the reciprocity law dramatically; Emma Lehmer writes
It is well known that the famous Legendre law of quadratic reciprocity, of which over 150 proofs are in print, has been generalized over the years to algebraic fields by a number of famous mathematicians from Gauss to Artin to the extent that it has become virtually unrecognizable.
(…)
In a way, Artin's reciprocity law closed the subject (except for the subsequent work on explicit formulas, not to mention the dramatic progress into non-abelian class field theory that is connected in particular with the names of Shimura and Langlands or the recent generalization of class field theory to “higher dimensional” local fields), and the decline of interest in the classical reciprocity laws was a natural consequence. Nevertheless, two of the papers that helped shape the research in number theory during the second half of this century directly referred to Gauss's work on biquadratic residues: first, there's Weil's paper from 1949 on equations over finite fields in which he announced the Weil Conjectures and which was inspired directly by reading Gauss:
In 1947, in Chicago, I felt bored and depressed, and, not knowing what to do, I started reading Gauss's two memoirs on biquadratic residues, which I had never read before. (…) This led me in turn to conjectures about varieties over finite fields, ...
namely the Weil Conjectures, now Deligne's theorem (see Chapter 10).
The other central theme in number theory during the last few decades came into being in two papers by Birch & Swinnerton-Dyer (…); in these papers, the quartic reciprocity plays a central role in checking some instances of their conjectures (…). As a matter of fact, even the explicit formulas of Artin-Hasse were resurrected (and generalized) by Iwasawa, Coates and Wiles in order to make progress on the Birch-Swinnerton-Dyer conjecture.»
BY Непрерывное математическое образование
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