# Integer-Valued Polynomials

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*Paul-Jean Cahen; Jean-Luc Chabert*

Integer-valued polynomials on the ring of integers have
been known for a long time and have been used in calculus. Pólya
and Ostrowski generalized this notion to rings of integers of
number fields. More generally still, one may consider a domain
\(D\) and the polynomials (with coefficients in its quotient
field) mapping \(D\) into itself. They form a
\(D\)-algebra—that is, a \(D\)-module with a ring
structure. Appearing in a very natural fashion, this ring possesses
quite a rich structure, and the very numerous questions it raises allow
a thorough exploration of commutative algebra. Here is the first book
devoted entirely to this topic.

Features:

- Thorough reviews of many published works.
- Self-contained text with complete proofs.
- Numerous exercises.

#### Reviews & Endorsements

Begins with two interesting introductions (both historical and mathematical) and includes almost 300 exercises … this makes the text not only a volume for experts, but usable in a classroom setting. Its bibliography … is by far the most extensive on this subject … an excellent book for readers new to the subject. For readers familiar with the field, it will be the key reference for many years to come.

-- Mathematical Reviews

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.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Integer-Valued Polynomials

- Contents vii8 free
- Foreword xi12 free
- Historical Introduction xiii14 free
- Mathematical Introduction xvii18 free
- Conventions and Notation xix20 free
- Chapter I. Coefficients and Values 122 free
- Chapter II. Additive Structure 2546
- Chapter III. Stone-Weierstrass 5172
- Chapter IV. Integer-Valued Polynomials on a Subset 7394
- Chapter V. Prime Ideals 99120
- Chapter VI. Multiplicative Properties 123144
- Chapter VII. Skolem Properties 159180
- Chapter VIII. Invertible Ideals and the Picard Group 193214
- Chapter IX. Integer-Valued Derivatives and Finite Differences 227248
- Chapter X. Integer-Valued Rational Functions 257278
- Chapter XI. Integer-Valued Polynomials in Several Indeterminates 285306
- References 307328
- List of Symbols 317338
- Index 319340