Reciprocity laws. From Euler to Eisenstein. (English) Zbl 0949.11002

Springer Monographs in Mathematics. Berlin: Springer. xix, 487 p. (2000).
This book is about reciprocity laws in number theory, their historical development and their connections with various parts of algebraic number theory, algebraic geometry and complex function theory. It consists of 11 chapters, three appendices, a list of 885 references, an author index and a subject index. Each chapter ends with a collection of “notes” followed by exercises and a list of additional references.
Chapter 1 describes the genesis of quadratic reciprocity and discusses mainly the corresponding work of Fermat, Euler, Legendre and Gauß. Especially it is shown how the problem of representing numbers by binary quadratic forms is related to quadratic reciprocity. The aim of Chapter 2 is to present proofs of the quadratic reciprocity law which are based on the theory of quadratic number fields. It contains a section on quadratic fields, on genus theory and genus characters. Moreover it discusses the Lucas-Lehmer primality test for Mersenne numbers, Hilbert symbols and describes the role of the second \(K\)-group of algebraic \(K\)-theory in quadratic reciprocity. Chapter 3 is devoted to some proofs of the quadratic reciprocity law which make use of the arithmetic of cyclotomic number fields. Primality tests which use the Artin symbol are also described. The “notes” for this chapter contain interesting historical remarks related to normal integral bases for subfields of cyclotomic fields. Chapter 4 discusses symbols in number fields, a generalization of the so-called Gauß-lemma for quadratic residues to general power residues, sections on discriminants of number fields, Kummer extensions, characters of finite abelian groups and on Gauß-, Jacobi- and Eisenstein-sums.
In Chapter 5 a detailed exposition of the so-called rational reciprocity laws, which deal with residue symbols assuming only the values \(+1\) or \(-1\) and which have all their entries in the rational integers, is given. The first reciprocity law of this type was discovered by Dirichlet, and later many variants and variations were published. Chapter 6 is about quartic reciprocity, discovered and proved by Gauß, and Chapter 7 discusses the cubic reciprocity law. In Chapter 8, one of the many highlights of this book, Eisenstein’s analytical proofs for quadratic, cubic and quartic residues are presented in full detail. Especially the role of elliptic functions in reciprocity is carefully explained. Chapter 9 is devoted to the octic reciprocity law.
Chapter 10 gives highly interesting discussion of the last entry of Gauß’s famous diary. As the author remarks, in present day terms this entry says the following: “Let \(p= a^2+b^2\equiv 1\bmod 4\) be a prime and suppose that \(a+ bi\equiv 1\pmod {2+2i}\). Then the congruence \(x^2+ y^2+ x^2y^2= 1\bmod p\) has exactly \(p-3-2a= (a-1)^2+ b^2-4\) solutions.” The author shows how this result is connected to biquadratic reciprocity, elliptic functions, elliptic curves and the Riemann hypothesis.
Chapter 11 is on Eisenstein reciprocity. In the introduction to this chapter the author remarks: “Although Eisenstein’s reciprocity law is only a very special case of more general reciprocity laws, it turned out to be an indispensable step for proving these general laws until Furtwängler … succeeded in finally giving a proof of the reciprocity law in \(\mathbb Q(\zeta_1)\) without the help of Eisenstein’s reciprocity law … Using the prime ideal factorization of Gauss sums together with the trivial fact that the \(m\)th power of Gauss sums generate principal ideals in \(\mathbb Z [\zeta_m]\), we will be able to deduce amazing properties of ideal class groups of abelian extensions of \(\mathbb Q\). This idea goes back to work of Jacobi, Cauchy and Kummer, was extended by Stickelberger and reviewed by Iwasawa. Later refinements and generalizations due to Thaine, Kolyvagin and Rubin will be discussed only marginally.”
It is obvious from this short account that the book covers an enormous amount of material. The author deserves our admiration for leading us through centuries of number theory up to present day research by following a unique Leitmotiv.
Some remarks: In connection with the role of the transfer map in quadratic reciprocity, mentioned on page 23, the interested reader might wish to consult also A. Weil’s famous paper “Sur la théorie de corps de classes” [J. Math. Soc. Japan 3, 1–35 (1951; Zbl 0044.02901)], in which the transfer map is used to establish the general Artin reciprocity isomorphism. A nice and concise description of the development of class field theory is also contained in the article “Dennis Garbanati, Class field theory summarized, Rocky Mt. J. Math. 11, 195–225 (1981; Zbl 0489.12004)”. Some beautiful ideas of T. Kubota seem to open a completely new approach to higher reciprocity [see especially his paper “Geometry of numbers and class field theory”, Jap. J. Math., New Ser. 13, 235–275 (1987; Zbl 0639.12004) and his overview “A foundation of class field theory applying properties of spatial figures”, Sugaku Exp. 8, 1–13 (1995; Zbl 0839.11054)]. In order to get an impression of Kubota’s ideas let us quote from Section 3 of the epilogue of this overview: “… As is realized by the fact that quadratic reciprocity was a major source of the remarkable development of classical number theory, the theory of power residues has an outstanding importance in number theory and is incomparably beautiful as well. The endeavor to complete the theory of power residues finally produced class field theory … in the process of the research, people were perhaps dominated by the belief that power residues and extension fields are mutually inseparable. This kind of belief has the possibility of being strengthened to a belief that number theory is the theory of extension fields, including nonabelian ones. If, however, one regards to reciprocity law as being an assertion within a field and tries to prove it without going through the extension fields, then, as explained in the present article, it is proved in a geometric way in terms of intuitive properties of spatial figures only by means of tools that already existed at the time of Euler …”
It might be interesting to compare these remarks with the first sentence of the book under review: “The history of reciprocity laws is a history of algebraic number theory”.
Editorial addendum: A table of errata is available at


11-03 History of number theory
11A15 Power residues, reciprocity
01-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to history and biography
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11R18 Cyclotomic extensions
11R37 Class field theory