Aside from its unquestionable novelty， leading to its inclusion in most if not all introductory courses in number theory， the law of quadratic reciprocity stands out as one of the deepest facts of the theory of algebraic number fields. This was certainly already understood by Gauss， who in his lifetime gave six proofs of this beautiful theorem first conjectured by Euler， There are a number of good sources available treating this central theme of Gauss‘ arithmetical work， among which we recommend Variationen uber ein Zahlentheoretisches Thema von Carl Friedrich Gauss [Pi78]， and the indicated section of Scharlau-Opolka [SO84].
Gauss’ work laid bare deep connections between at first glance rather disparate aspects of the behavior of rings of integers of algebraic number fields. Presently it became clear that the splitting of primes in quadratic extensions is completely governed by the fine structure of the Legendre symbol， that is， by quadratic reciprocity， and this set the stage for Gauss‘ work on the genera of quadratic forms.
If there is a tool par excellence in Gauss’ armory for these arithmetical investigations it is surely the method of Gauss sums. Their relation to the Legendre symbol is fundamental； it is an easy exercise to show that Gauss sums transform ver）r nicely under the Legendre symbols natural action. It is a quick step from there to the formulation of quadratic reciprocity as an identity between so-called reciprocal Gausssums. But where are the quadratic forms?
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This is not a book for experts. This is not a book for raw beginners. It is， instead， an exposition of and commentary on a handful of sources， most of them classical by now and at least one of them notoriously austere. The material presented has been compiled as an aid to number theorists seeking to work on the analytic proof of reciprocity laws. This is a notorious affair， of course. The quadratic case is completely settled by Hecke [He23]， and resettled by Weil [We64J， but for n＞2 the matter is still open and ranks as one of the hardest open problems in the field. This book is written for those reckless few who are predisposed to enter this area of research at an early （but not too early） stage of their career， when they don‘t yet know any better and don’t know a lot about the indicated specialized techniques either. The goal is to make entry into this field a little easier by explicitly delineating and comparing the three existing approaches to the （Fourier-） analytic proof of quadratic reciprocity which qualifies as a paradigm for the general case.
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