\(p\)-adic analysis compared with real.

*(English)*Zbl 1147.12003
Student Mathematical Library 37. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4220-1/pbk). xiii, 152 p. (2007).

The book under review is a nice and well-thought out introduction to \(p\)-adic numbers. It is based on an advanced undergraduate course given by the author in the Mathematics Advanced Study Semesters Program.

The main protagonist is the field \(\mathbb{Q}_p\) of the \(p\)-adic numbers, where \(p\) is a prime number. This field is the root of the \(p\)-adic (or non-archimedean) world and at the same time one of the most important examples of non-archimedean valued fields, because of its influence in the applications. \(p\)-adic numbers appear in all the branches of non-archimedean mathematics. In the present book the author centers her attention in three of the most relevant ones: Number Theory, Topology and Analysis.

I think that the words “nice” and “well-thought out” used at the beginning of this review are appropriate to “judge” the book. I explain below some of the reasons that lead to me to use them.

\(\bullet\) The choice of the topics treated in the book: Arithmetic of the \(p\)-adic numbers (Chapter 1); The topology of \(\mathbb{Q}_p\) vs. the topology of \(\mathbb{R}\) (Chapter 2); Elementary Analysis in \(\mathbb{Q}_p\) (Chapter 3); \(p\)-adic functions (Chapter 4). They reveal the internal beauty of the \(p\)-adic mathematics, an unusual subject in the undergraduate curriculum.

\(\bullet\) The emphasis on comparing the non-archimedean situation with the much more familiar real (or classical) counterpart.

Both, real and \(p\)-adic numbers, are obtained by completing the field of rational numbers, by using different valuations: the usual absolute value (already known by an undergraduate student) and the \(p\)-adic valuation respectively. This construction of \(\mathbb{Q}_p\) from the rationals, as well as its algebraic and structural properties, are studied in Chapter 1.

Further, it is shown in the book that the strong triangle inequality of the \(p\)-adic valuation is responsible of properties that are surprising for an archimedean mind and that imply sharp (and interesting) deviations from the classical case. I point out the following: The balls in \(\mathbb{Q}_p\) are open and closed (in the topological sense), so \(\mathbb{Q}_p\) is totally disconnected (Chapter 2); a series in \(\mathbb{Q}_p\) converges if and only if its general term tends to \(0\), a \(p\)-adic power series cannot be analytically continued, the formal power series that leads to the exponential function only converges in a disc in \(\mathbb{Q}_p\) of center \(0\) and radius \(<1\), so we do not have a \(p\)-adic number “\(e\)” (Chapter 3); there exist non-constant functions that are locally constant and also injective functions whose derivatives are \(0\) (Chapter 4).

This work of comparison developed by the author reveals an extra beauty of the \(p\)-adic mathematics apart from its internal one mentioned above. This work also allows the students to better understand the proofs of results in both contexts.

\(\bullet\) The inclusion of a large number of exercises, which are very adequate for the people to whom I think this book is written: advanced undergraduate or postgraduate students who want to learn the basics about \(p\)-adic numbers. In fact, several of these exercises consist on proofs and in some cases even proofs of results appearing in the main body of the text.

This inclusion is a good idea because, on the one hand, it helps student\('\)s deeper understanding of the subjects and, on the other hand, it “forces” students to carry out by themselves serious mathematical reasonings. The last is in general a hard task during the first courses of the degree. But now they are advanced undergraduate or postgraduate students and it is the moment to practice the “art of the reasoning”. This book gives to them a good opportunity to do and enjoy this art. Anyway, at the end of the book there are hints for most of the exercises.

I think that the reading of this book could animate some students to start doing research \(p\)-adic work. A good decision from my point of view!

The main protagonist is the field \(\mathbb{Q}_p\) of the \(p\)-adic numbers, where \(p\) is a prime number. This field is the root of the \(p\)-adic (or non-archimedean) world and at the same time one of the most important examples of non-archimedean valued fields, because of its influence in the applications. \(p\)-adic numbers appear in all the branches of non-archimedean mathematics. In the present book the author centers her attention in three of the most relevant ones: Number Theory, Topology and Analysis.

I think that the words “nice” and “well-thought out” used at the beginning of this review are appropriate to “judge” the book. I explain below some of the reasons that lead to me to use them.

\(\bullet\) The choice of the topics treated in the book: Arithmetic of the \(p\)-adic numbers (Chapter 1); The topology of \(\mathbb{Q}_p\) vs. the topology of \(\mathbb{R}\) (Chapter 2); Elementary Analysis in \(\mathbb{Q}_p\) (Chapter 3); \(p\)-adic functions (Chapter 4). They reveal the internal beauty of the \(p\)-adic mathematics, an unusual subject in the undergraduate curriculum.

\(\bullet\) The emphasis on comparing the non-archimedean situation with the much more familiar real (or classical) counterpart.

Both, real and \(p\)-adic numbers, are obtained by completing the field of rational numbers, by using different valuations: the usual absolute value (already known by an undergraduate student) and the \(p\)-adic valuation respectively. This construction of \(\mathbb{Q}_p\) from the rationals, as well as its algebraic and structural properties, are studied in Chapter 1.

Further, it is shown in the book that the strong triangle inequality of the \(p\)-adic valuation is responsible of properties that are surprising for an archimedean mind and that imply sharp (and interesting) deviations from the classical case. I point out the following: The balls in \(\mathbb{Q}_p\) are open and closed (in the topological sense), so \(\mathbb{Q}_p\) is totally disconnected (Chapter 2); a series in \(\mathbb{Q}_p\) converges if and only if its general term tends to \(0\), a \(p\)-adic power series cannot be analytically continued, the formal power series that leads to the exponential function only converges in a disc in \(\mathbb{Q}_p\) of center \(0\) and radius \(<1\), so we do not have a \(p\)-adic number “\(e\)” (Chapter 3); there exist non-constant functions that are locally constant and also injective functions whose derivatives are \(0\) (Chapter 4).

This work of comparison developed by the author reveals an extra beauty of the \(p\)-adic mathematics apart from its internal one mentioned above. This work also allows the students to better understand the proofs of results in both contexts.

\(\bullet\) The inclusion of a large number of exercises, which are very adequate for the people to whom I think this book is written: advanced undergraduate or postgraduate students who want to learn the basics about \(p\)-adic numbers. In fact, several of these exercises consist on proofs and in some cases even proofs of results appearing in the main body of the text.

This inclusion is a good idea because, on the one hand, it helps student\('\)s deeper understanding of the subjects and, on the other hand, it “forces” students to carry out by themselves serious mathematical reasonings. The last is in general a hard task during the first courses of the degree. But now they are advanced undergraduate or postgraduate students and it is the moment to practice the “art of the reasoning”. This book gives to them a good opportunity to do and enjoy this art. Anyway, at the end of the book there are hints for most of the exercises.

I think that the reading of this book could animate some students to start doing research \(p\)-adic work. A good decision from my point of view!

Reviewer: Cristina Pérez-García (Santander)

##### MSC:

12J25 | Non-Archimedean valued fields |

46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |

46-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis |

12-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory |

11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |

26E30 | Non-Archimedean analysis |

30G06 | Non-Archimedean function theory |