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Divisor theory. (English) Zbl 0689.12001
Boston etc.: Birkhäuser Verlag. xiv, 166 p. DM 68.00; sFr. 58.00 (1990).

The second half of the 19th century was a period during which were elaborated man algebraic theories and techniques. On of the most reknown was the invention of the ideals by Richard Dedekind, because of the abundant subsequent developments and applications [cf. H. M. Edwards, Arch. Hist. Exact. Sci. 23, 321-378 (1980; Zbl 0472.01013)]. But some of these theories and techniques remained forgotten for many years, even if susceptible of fruitfull applications.

One of these theories is that of the algebraic divisors and was described by Leopold Kronecker in a longer paper published in Crelle in 1882. It was considered an alternative introduction of the ideals. In the case of the polynomials with coefficients in an algebraic number field K that contains $ℚ$ the multiplicative group of the algebraic divisors coincides with the group of the Dedekind ideals in K. In the first decades after the publication of Kronecker’s paper the divisor theory was considered with suspecion by some algebraists. For instance, J. König [in "Einleitung in die allgemeine Theorie der algebraischen Größe$\parallel |$ Teubner (1903), p. 483] formulated the opinion that it is inconvenient and not suitable for applications. Contrary to the opinion of this contemporary of Kronecker, the author of this book illustrates the consistency of the theory of the algebraic divisors as introduced in the paper of 1882. He presents the motivation of the theory and its sources in Gauss’ lemma about the content of a product of two polynomals in one variable with integer coefficients. The initial divisor theory is exposed with several welcome simplifications and extensions. Some advantages with respect to Dedekind’s theory are pointed, especially those arising in algorithmic problems. Many substantial applications to algebraic number theory and to algebraic curves are presented, such as a method for the factorization into primes in an algebraic number field, Dedekind’s discriminant theorem, the study of the quadratic reciprocity, the places and the genus of a function field, a new version of the theorem of Abel about Abelian integrals. The text also includes an appendix about the differentials in a function field.

The book of H. M. Edwards presents in a modern approach a remarkable algebraic technique. It is written in such a matter that it is interesting for both the historian of mathematics and the working specialist in commutative algebra, number theory and algebraic geometry.

##### MSC:
 12-03 Historical (field theory) 13-03 Historical (commutative algebra) 01A55 Mathematics in the 19th century 13A15 Ideals; multiplicative ideal theory 12E05 Polynomials over general fields 13B25 Polynomials over commutative rings