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Integer-valued polynomials. (English) Zbl 0884.13010
Mathematical Surveys and Monographs. 48. Providence, RI: American Mathematical Society (AMS). xviii, 322 p. (1997).
This is the first book entirely devoted to the study of algebraic properties of the ring \(\text{Int} (R)\) of polynomials which map a commutative domain \(R\) into \(R\). The case when \(R\) is the ring of integers of an algebraic number field was first studied by Pólya and Ostrowski in the twenties, and during the last 25 years several authors have studied various aspects of the general case, a large part of the work being done by the present authors. – After some preliminary results, the additive structure of \(\text{Int} (R)\) is studied and in particular a necessary and sufficient condition for the existence of a sequence \(f_n\) of polynomials with \(\deg f_n=n\) which generates \(\text{Int} (R)\) as an \(R\)-module (regular basis) is proved. The case of a Dedekind domain \(R\) is treated in more detail.
In the next chapter the question of the density of \(\text{Int} (D)\) in the space of continuous functions on \(\widehat D\), the completion of an infinite noetherian domain \(D\), is considered (Stone-Weierstrass approximation theorem). Then, in chapter IV, the ring \(\text{Int} (E,R)\) of polynomials which are \(R\)-valued on a fixed subset \(E\) of the cluotient field of \(R\) is studied. In chapter V prime ideals of \(\text{Int} (D)\) (with \(D\) an infinite noetherian domain) are described. The next chapter is devoted to multiplicative properties of \(\text{Int} (R)\) and in particular it is proved that if \(R\) is a Dedekind domain with the finite norm property (i.e. all non-zero ideals are of finite index) then \(\text{Int} (R)\) is a two-dimensional Prüfer domain.
In 1936 Th. Skolem observed that if the values of \(f_1,f_2,\dots, f_n\in \text{Int} (\mathbb{Z})\) at every integer are relatively prime, then these polynomials generate the unit ideal. In chapter VII the analogue of this property [introduced by D. Brizolis, Commun. Algebra 3, 1051-1081 (1975; Zbl 0318.13009) and called by him the Skolem property] is considered and infinite noetherian domains \(D\) for which \(\text{Int} (D)\) have this property are characterized. In the next two chapters the Picard group of \(\text{Int} (R)\) and polynomials whose differences or derivatives lie in \(\text{Int} (R)\) are studied and the final chapters deal with generalizations to rational functions and polynomials in several variables. – At the end of each chapter one finds a large choice of interesting (but not always easy) exercises.
The authors succeeded in presenting everything of importance in the theory of integer-valued polynomials and this short review cannot do justice to the rich contents of their book. The presentation of the material is very good and the book offers a pleasant reading.

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13B25 Polynomials over commutative rings
11C08 Polynomials in number theory
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra
11-02 Research exposition (monographs, survey articles) pertaining to number theory
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
11R09 Polynomials (irreducibility, etc.)
13A15 Ideals and multiplicative ideal theory in commutative rings
13G05 Integral domains
13C20 Class groups
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13E05 Commutative Noetherian rings and modules