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Classical theory of algebraic numbers. (English) Zbl 1082.11065
Universitext. New York, NY: Springer (ISBN 0-387-95070-2/hbk). xxiv, 681 p. (2001).
Ribenboim’s ‘Classical Theory of Algebraic Numbers’ is an introduction to algebraic number theory on an elementary level, assuming only basic algebra, Galois theory, and real analysis. The first sixteen chapters, which comprise about half the book, are an improved version of the author’s first book [Algebraic numbers, New York, Wiley-Interscience, 1972; Zbl 0247.12002] and cover the standard material: algebraic integers, discriminants, decomposition of prime ideals, finiteness of the class number, Dirichlet’s unit theorem, Hilbert’s ramification theory and the theorem of Kronecker-Weber.
The second half of the book is new, consists of an algebraic and an analytic part, and is more or less centered around the proof of Fermat’s Last Theorem for regular primes.
Chapters 17 and 18 introduce just enough theory of local fields (basically just the completions of \(\mathbb Q(\zeta_p)\)) and Bernoulli numbers as is needed for a proof of Fermat’s Last Theorem for regular primes \(p\) in Chapter 19; Kummer’s logarithmic derivatives that he used in his proof of Kummer’s Lemma are replaced by arguments involving \(p\)-adic logarithms. Chapter 20 contains results on the algebraic solution of the cyclotomic equation, Gaussian periods, ‘Lagrange resolvents’ and ‘Jacobi’s cyclotomic functions’ (essentially Gauss and Jacobi sums), and gives a proof of Kummer’s result that the class groups of cyclotomic fields are generated by prime ideals of degree \(1\).
After a chapter on characters and Gauss sums, zeta functions \(\zeta(s)\) and L-series \(L(s,\chi)\) are introduced (for real values of \(s\)) and used to prove Dirichlet’s theorem on primes in arithmetic progressions, class number formulas for quadratic and cyclotomic fields, and Chebotarev’s density theorem. This is complemented by a final chapter containing miscellaneous results on class numbers of cyclotomic fields.
The number of exercises ranges from about 50 for the first few chapters down to four for Chapter 27; most of them are standard calculations, others fill in details of proofs or introduce new concepts. The presentation of the material is clear and well motivated, and the book contains many (too many?) examples.
There are a few points that the reviewer is not particularly happy about. One is the lack of references to original papers: for example when discussing the equations \(x^2 - dy^2 = \pm 1, \pm 4\) on p. 173, the author says that “they were studied in the textbook by Pell”.
Another point concerns Theorem 2 in Chapter 11, which gives the prime ideal decomposition of primes \(p\) in number fields \(K\) in terms of the factorization modulo \(p\) of the generating polynomial of \(K\) only for monogenic rings or, in special cases, for primes not dividing the index, and this gap is not closed in the chapter on local fields. On p. 294 the author then gives an algorithm that he claims can compute the decomposition of prime ideals in finitely many steps, but for a proof he is referring back to the incomplete theorem mentioned above.
The discussion of the main theorems on Hilbert class fields (without proofs) suffers from the defect that ramification at infinite primes is not discussed properly. This leads to a small gap in the proof that the class number of \(K\) is even if there is an extension \(L/K\) with relative discriminant \(1\) on p. 621, and to a weakened statement of the principal ideal theorem (all ideals in the usual sense become principal in the absolute class field, which is the author’s name for the Hilbert class field in the strict sense).
The ideal theoretic description of class field theory is complete, with the exception of Artin’s reciprocity law “which is conveniently stated appealing to \([\ldots]\) idèles”. This is a bit strange in light of the fact that the Frobenius symbol is introduced and discussed in the chapter on Chebotarev’s density theorem, and that Chebotarev’s paper contained the key idea for Artin’s proof of his reciprocity law.
Summing up, Ribenboim’s book is a well written introduction to classical algebraic number theory as contained e.g. in Hilbert’s Zahlbericht [see Zbl 0977.11500], and the perfect textbook for students who need lots of examples.

11Rxx Algebraic number theory: global fields
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory