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Primes of the form \(x^ 2+ny^ 2\). Fermat, class field theory and complex multiplication. (English) Zbl 0701.11001
New York etc.: John Wiley & Sons. xi, 351 p. £34.45/hbk, $ 49.95/pbk (1989).
The author unifies algebraic number theory (in particular, Weber’s version of class field theory) by concentrating on the problem associated with Fermat, namely the conditions for representing the prime \(p\) by \(x^ 2+ny^ 2.\) For small values of \(n\), the conditions are congruence classes for \(p\) (modulo \(4n\), for example). As \(n\) becomes larger (say \(n=14\)), the problem is transmuted into a gigantic mathematical adventure, namely the search for special (class) polynomials of \(\mathbb Z[x]\), the generators of class fields over \(\mathbb Q(\sqrt{-n}).\) (For \(n=14\), the class polynomial is \((x^ 2+1)^ 2-8.)\) These polynomials have the property that the representability of \(p\) reduces to the splitting of such polynomials into linear factors in \(x\) modulo \(p\). (The “reciprocal” relation of \(4n\) and \(p\) is not permitted to be lost on the reader, with Artin reciprocity). The next big step is the introduction of modular equations to carry out the search for class polynomials, or complex multiplication. (Here as a special feature, the interrelation of the theorems of M. Deuring [Comment. Math. Helv. 19, 74–82 (1946; Zbl 0061.063)] and B. Gross and D. Zagier [J. Reine Angew. Math. 355, 191–220 (1985; Zbl 0545.10015)] on discriminants of singular moduli are discussed.) The adventure ends in a very contemporary fashion with the study of efficient primality tests, of the Lenstra elliptic curve type [see H. Lenstra, Ann. Math. (2) 126, 649–673 (1987; Zbl 0629.10006)]. The author also discusses the Goldwasser-Killian-Atkin test (and in so doing he might be providing the best available current reference).
The book is very carefully documented, but with considerable restraint, as the reader is not forced to follow the unproductive historical meanderings. The more difficult proofs are referred elsewhere [e.g., S. Lang, Algebraic number theory (1986; Zbl 0601.12001) and Elliptic functions (1987; Zbl 0615.14018)] but the presentation is nevertheless very convincing, due to the presence of a detailed exposition with many supporting problems. The reviewer’s books [A classical invitation to algebraic numbers and class fields (1978; Zbl 0395.12001) and Introduction to the construction of class fields (1985; Zbl 0571.12001)] had a shorter (and less current) version of the quadratic form approach, but in which Riemann surfaces and Hilbert’s class field approach (by principalization and radicals) were also discussed. Other historical works which the author cites especially are A. Weil [Number theory. An approach through history (1984; Zbl 0531.10001)] and W. Scharlau and H. Opolka [From Fermat to Minkowski (1985; Zbl 0426.10001)]. The author’s version makes the incredible jump from forms to elliptic prime-testing seem almost preordained. In short, this is a very hard book for the reader to put back on the shelf unfinished.

11-02 Research exposition (monographs, survey articles) pertaining to number theory
11R37 Class field theory
11A51 Factorization; primality
11G15 Complex multiplication and moduli of abelian varieties
11Y11 Primality
11E25 Sums of squares and representations by other particular quadratic forms