## Reviews of Igor Shafarevich's books

**1. On the solution of equations of higher degrees (Russian) (1954).**

**1.1. Review by: Anon.**

In the series, 'Popular lectures on mathematics'.

**2. The theory of numbers (with Zenon Ivanovich Borevich) (Russian 1964, English 1966).**

**2.1. From the Preface.**

This book is written for the student in mathematics. Its goal is to give a view of he theory of numbers, of the problems with which this theory deals, and of the methods that are used. We have avoided that style which gives a systematic development of the apparatus and have used instead a freer style, in which the problems and the methods of solution are closely interwoven. We start from concrete problems in number theory. General theories arise as tools for solving these problems. As a rule, these theories are developed sufficiently far so that the reader can see for himself their strength and beauty, and so that he learns to apply them.

**2.2. Review by: Sarvadaman Chowla.**

*Mathematical Reviews*, MR0170845 **(30 #1080)**.

This is a beautifully written work. ... Chapter 1: Congruences. ... Chapter 2 contains a discussion of norm-forms, Minkowski's Lemma, Dirichlet's Theorem on the structure of the group of units, and the conditions for a reduced binary quadratic form of negative discriminant. Chapter 3 is on divisibility and divisors. Amongst items discussed here are an introduction to the cyclotomic field and Fermat's Last Theorem, the euclidean algorithm in real (no reference here to Davenport and Chatland) and imaginary quadratic fields. Also discussed are class-numbers of algebraic number-fields. ... Chapter 4 is concerned with "local methods". ... Chapter 5 is called "analytic methods".

**2.3. Review by: Ivan Niven.**

*Amer. Math. Monthly* **74** (6) (1967), 751.

This is a quite distinctive and valuable book because it overlaps very little the other available texts on the subject in English. Written with verve and sophistication, it offers a high-level second course on the subject. ... the topics discussed come under the headings of congruences, divisibility, and Diophantine equations, especially representation by forms. It should be added at once that these are treated in a broad framework going far beyond the discussion of integers or rational numbers.

**2.4. Review of 3rd edition published 1985: by Jeffrey S Joel.**

*Mathematical Reviews*, MR0816135 **(88f:11001)**.

... the reviewer would like to note that the book wears its age very lightly. The topics and treatment are so current that it is hard to believe the book was written nearly 25 years ago.

**3. Lectures on minimal models and birational transformations of two dimensional schemes (1966).**

**3.1. From the Preface.**

These lectures contain an exposition of fundamental concepts and results of the theory of birational transformations and minimal models for schemes of dimension two. In the case of surfaces over an algebraically closed field of characteristic zero, these results were obtained by old italian geometers. In the case of fields of arbitrary characteristic, they are due to Zariski (cf. his book on minimal models). Later Neron observed the importance of obtaining such results in the case of schemes of dimension 2, for certain questions of number theory. In particular, he proved the existence of absolute minimal models for two dimensional schemes over rings of integers of global fields in the case where the genus of the generic fibre is 1. The first aim of these lectures was to give a proof of the corresponding result in the case of arbitrary genus g.

**3.2. Review by: Joseph Lipman.**

*Mathematical Reviews*, MR0217068 **(36 #163)**.

These eight lectures are devoted to a number of basic topics in the theory of surfaces (i.e., Noetherian schemes all of whose components are two-dimensional), leading up to the problem of minimal models over a Noetherian base scheme ... The lectures are enhanced by many examples - the treatment of the twenty-seven lines on a cubic surface realized by blowing up six points in the projective plane, and Nagata's example of a non-projective surface, to mention two.

**4. Basic algebraic geometry (Russian 1972, English 1974).**

**4.1. From the Translator's Note.**

Shafarevich's book is the fruit of the lecture courses at Moscow State University in the 1960s and early 1970s. The style of Russian mathematical writing of the period is very much in evidence. The book does not aim to cover a huge volume of material in the maximal generality and rigour, but gives instead a well-considered choice of topics, with a human-oriented discussion of the motivation and the ideas, and some sample results (including a good number of hard theorems with complete proofs).

**4.2. Review by: Peter E Newstead.**

*Mathematical Reviews*, MR0366916 **(51 #3162)**.

This book is an account of the elements of algebraic geometry based on courses given by the author at Moscow University. It is divided into three main parts, which differ somewhat from one another in their approach. Part I (Chapters I-IV) consists of a detailed exposition of the theory of quasi-projective varieties, which has already appeared in another form ...; there are a few minor revisions, but for details the reader should see the original review. Part II (Chapters V and VI) is a brief introduction to the theory of schemes; the author concentrates here on definitions and examples, and shows how the basic concept of a variety can be recovered and perhaps more fully understood in this context. Part III (Chapters VII-IX) is a survey of the basic topological and analytic properties of varieties defined over the complex numbers. There is also a brief historical sketch.

**5. Geometries and groups (with Viacheslav Valentinovich Nikulin) (Russian 1983, English 1987).**

**5.1. From the Preface.**

This book is devoted to the theory of geometries which are locally Euclidean, in the sense that in small regions they are identical to the geometry of the Euclidean plane or Euclidean 3-space. Starting from the simplest examples, we proceed to develop a general theory of such geometries, based on their relation with discrete groups of motions of the Euclidean plane or 3-space; we also consider the relation between discrete groups of motions and crystallography. The description of locally Euclidean geometries of one type shows that these geometries are themselves naturally represented as the points of a new geometry. The systematic study of this new geometry leads us to 2-dimensional Lobachevsky geometry (also called non-Euclidean or hyperbolic geometry) which, following the logic of our study, is constructed starting from the properties of its group of motions. Thus in this book we would like to introduce the reader to a theory of geometries which are different from the usual Euclidean geometry of the plane and 3-space, in terms of examples which are accessible to a concrete and intuitive study. The basic method of study is the use of groups of motions, both discrete groups and the groups of motions of geometries. The book does not presuppose on the part of the reader any preliminary knowledge outside the limits of a school geometry course.

**5.2. Review by: Karl Strambach.**

*Mathematical Reviews*, MR0742693 **(86f:51021)**.

This book mainly deals with 2- and 3-dimensional locally Euclidean geometries; these are metric geometries realized on a manifold, so that for every point there exists a neighbourhood which is isometrically isomorphic to the unit disc [resp. unit ball] in 2-dimensional [resp. 3-dimensional] Euclidean space. ... The historical remarks in the book stand out, in our opinion, through exactness and, especially, fairness. Since the book is certain to have a broad audience, it certainly will be translated into English. Nevertheless we would recommend the original to many, for we feel the Russian of the authors to be very graphic and expressive (although our knowledge of Russian is limited).

**5.3. Review by: Heinrich W Guggenheimer.**

*Amer. Math. Monthly* **96** (4) (1989), 370-373.

The book of Nikulin and Shafarevich gives a very careful, complete, and leisurely introduction to the results of Klein and Poincare on the Clifford-Klein space problem in two and three dimensions and of Jordan-(Fedorov-Schönfliess) in the plane, with attention to some aspects not covered in this historical essay, such as the problem of moduli of the flat torus which opens up vistas of algebraic geometry. The book is well written and edited (I found only 3 typos). It would serve admirably for a one-semester geometry course in a liberal arts college since its prerequisite is only a moderate amount of high school geometry. All topology and algebra is developed in the text. Even complex numbers are developed from first geometric principles.

**5.4. Review by: Hugh Cowie Williams.**

*The Mathematical Gazette* **73** (465) (1989), 257-258.

This book will be of interest to those who want to know about the classification of geometries and how this is related to certain types of groups. The book starts off by looking at the geometries on a cylinder, a sphere, a Mobius strip and a torus. All of these are shown to be locally Euclidean which roughly means that if points are near enough together they behave like points on the ordinary Euclidean plane. It is shown how each of these finite surfaces can be mapped onto the plane so that the surface has multiple images on the plane. As these multiple images are equivalent the group of mappings which sends one image into its equivalent images is very important. For the four examples these are all groups of the type known as uniformly discontinuous groups. ... The back cover expects this book to find a place alongside classics such as H Weyl's *Symmetry*. I do not agree. Basically it is too hard mathematics delivered too rigorously for it to have the breadth of appeal. It is an interesting book, well-written and translated but it takes a lot of hard work to get on top of it. It is not a book that can be dipped into but anyone who is interested in this area will find the effort in reading it rewarded.

**6. Basic notions of algebra (Russian 1986, English 1990).**

**6.1. Review by: Paul Moritz Cohn.**

*Mathematical Reviews*, MR0895587

**(88i:00007)**.

This book is a survey of algebra, emphasizing concepts, ideas, applications, and omitting all but the simplest proofs. The author devotes about 100 pages each to rings (including fields) and groups, and 50 to homological algebra; he has in mind a reader who is fairly ignorant in algebra but well versed in the rest of mathematics including theoretical physics. Thus rings are illustrated by function rings and rings of differential operators, quaternions are applied to describe the Hopf bifurcation, and Clifford algebras are invoked to factorize the Dirac equation. Groups are illustrated by simple examples, the symmetry groups of polyhedra and various crystals, but also by more abstract cases such as the Brauer group. ... The author has accomplished the seemingly impossible task of giving a readable survey of a vast area, which should encourage algebra users to delve deeper into the subject.

**6.2. Review by: Nick Lord.**

*The Mathematical Gazette* **75** (471) (1991), 120-121.

"This book makes no pretence to teach algebra; it is merely an attempt to talk about it. ... What is algebra? Is it a branch of mathematics, a method or a frame of mind?" These remarks, from the beginning of the book under review by the distinguished Russian mathematician I R Shafarevich, clearly set it apart as something very unusual. The author's aim is to provide a systematic survey of present-day algebraic notions and theories built around a framework of key examples (many reflecting Shafarevich's own interests in number theory and algebraic geometry) and applications of algebra (within mathematics and within science) with a view loosely to uphold his thesis that (p 8): "Anything which is the object of mathematical study ... can be "coordinatised" or "measured". However, for such a coordinatisation the "ordinary" numbers are by no means adequate. Conversely, when we meet a new type of object, we are forced to construct (or to discover) new types of "quantities" to coordinatise them. The construction and the study of the quantities arising in this way is what characterises the place of algebra in mathematics. ..." There are few proofs in full, but there is an exhilarating combination of sureness of foot and lightness of touch in the exposition (faithfully reflected in Miles Reid's translation) which transports the reader effortlessly across the whole spectrum of algebra: from fields, rings, modules to finite geometries and Lie algebras; from finite groups to Lie and algebraic groups; from representation theory to homological algebra and K-theory. ... The challenge to Ezekiel, "Can these bones live?" is, all too often, the reaction of students when introduced to the bare bones of the concepts and constructs of modem algebra. Shafarevich's book - which reads as comfortably as an extended essay - breathes life into the skeleton and will be of interest to many classes of readers; certainly beginning postgraduate students would gain a most valuable perspective from it but also, as a unique work of high-level popularisation, both the adventurous undergraduate and the established professional mathematician will find a lot to enjoy within its pages and all would charge the author with undue modesty in the quotation at the start of this review.

**7. Basic algebraic geometry. 1. Varieties in projective space (Russian 1988, English 1994).**

This book is a second edition of Basic algebraic geometry (Russian 1972, English 1974). However, this edition is in two volumes (the original was only one) and we list it separately.

**7.1. From the Preface.**

The first edition of this book came out just as the apparatus of algebraic geometry was reaching a stage that permitted a lucid and concise account of the foundations of the subject. The author was no longer forced into the painful choice between sacrificing rigour of exposition or overloading the clear geometrical picture with cumbersome algebraic apparatus. The 15 years that have elapsed since the first edition have seen the appearance of many beautiful books treating various branches of algebraic geometry. However, as far as I know, no other author has been attracted to the aim which this book set itself: to give an overall view of the many varied aspects of algebraic geometry, without going too far afield into the different theories. There is thus scope for a second edition. In preparing this, I have included some additional material, rather varied in nature, and have made some small cuts, but the general character of the book remains unchanged.

**7.2. Review by: Werner Kleinert.**

*Mathematical Reviews*, MR0969372 **(90g:14001)**.

The first edition of this book appeared (in Russian) in 1972. At that time, this textbook was the first and the only one which built bridges between the geometric notions, the classical origins and achievements, the modern concepts and methods, and the complex-analytic aspects in algebraic geometry. The English translation of this outstanding textbook was published in 1977 under the title *Basic algebraic geometry*. In the meantime, it has become one of the most valuable, recommended and used textbooks on algebraic geometry worldwide, together with the other standard textbooks of R Hartshorne, D Mumford, and P A Griffiths and J Harris. The special feature of the author's book, in comparison to the other textbooks, is provided by the fact that it really conveys the many different aspects of contemporary algebraic geometry, without focussing on any particular approach and without requiring any advanced prerequisites. In this sense, it has proved an extremely useful addition to the other (here and there) more thorough-going textbooks, a recommendable introduction to them and to current research, and in any case, an excellent invitation to algebraic geometry. Now the second edition of this celebrated textbook has appeared, this time in two volumes. The author maintains the tried and true arrangement of the first edition, i.e., he has left the aims, the character, and the chapters basically intact. However, taking into account further developments and interconnections of algebraic geometry during the past two decades, he has added (in an organic manner) some important topics of current interest as well as some further motivating and instructive examples.

**8. Basic algebraic geometry. 2. Schemes and complex manifolds (Russian 1988, English 1994).**

This book is a second edition of Basic algebraic geometry (Russian 1972, English 1974). However, this edition is in two volumes (the original was only one) and we list it separately.

**8.1. From the Preface.**

See 7.1 above.

**8.2. Review by: Werner Kleinert.**

*Mathematical Reviews*, MR0969764 **(90g:14002)**.

The second volume of the new edition of this textbook is an expanded version of Chapters V-IX of the first edition [1972]. Accordingly, it is devoted to the foundations of the theory of schemes and the theory of complex algebraic varieties. As in the first volume, the author enriches the original material by some additional and important topics, leaving the well-established disposition of the original version otherwise intact. ... As for this second volume, one can only respectfully repeat what has been said about the first part: a great textbook, written by a great mathematician, has been reworked and, in effect, has become even greater. Students, teachers, and active researchers in algebraic geometry, complex analysis, and mathematical physics will be grateful to the author for his service to the mathematical community.

**9. Discourses on algebra (2003).**

**9.1. From the Preface:**

I wish that algebra would be the Cinderella of our story. In the mathematics program in schools, geometry has often been the favorite daughter. The amount of geometric knowledge studied in schools is approximately equal to the level achieved in ancient Greece and summarized by Euclid in his *Elements* (third century B.C.). For a long time, geometry was taught according to Euclid; simplified variants have recently appeared. In spite of all the changes introduced in geometry courses, geometry retains the influence of Euclid and the inclination of the grandiose scientific revolution that occurred in Greece. More than once I have met a person who said, 'I didn't choose math as my profession, but I'll never forget the beauty of the elegant edifice built in geometry with its strict deduction of more and more complicated propositions, all beginning from the very simplest, most obvious statements!' Unfortunately, I have never heard a similar assessment concerning algebra. Algebra courses in schools comprise a strange mixture of useful rules, logical judgments, and exercises in using aids such as tables of logarithms and pocket calculators. Such a course is closer in spirit to the brand of mathematics developed in ancient Egypt and Babylon than to the line of development that appeared in ancient Greece and then continued from the Renaissance in western Europe. Nevertheless, algebra is just as fundamental, just as deep, and just as beautiful as geometry. Moreover, from the standpoint of the modern division of mathematics into branches, the algebra courses in schools include elements from several branches: algebra, number theory, combinatorics, and a bit of probability theory. The task of this book is to show algebra as a branch of mathematics based on materials closely bordering the course in schools. The book does not claim to be a textbook, although it is addressed to students and teachers. The development presumes a rather small base of knowledge: operations with integers and fractions, square roots, opening parentheses and other operations on expressions involving letter symbols, the properties of inequalities. All these skills are learned by the 9th grade. The complexity of the mathematical considerations increases somewhat as we move through the book. To help the reader grasp the material, simple problems are given to be solved. The material is grouped into three basic themes - numbers, polynomials, and sets - each of which is developed in several chapters that alternate with the chapters devoted to the other themes. Certain matters related to the basic text, although they do not use more ideas than are already present, are more complicated and require that the reader keep more facts and definitions in mind. These matters are placed in supplements to the chapters and are not used in subsequent chapters. For the proofs of assertions given in the book, I chose not the shortest but the most 'understandable'. They are understandable in the sense that they connect the assertion to be proved with a larger number of concepts and other assertions; they thus clarify the position of the assertion to be proved within the structure of the presented area of mathematics. A shorter proof often appears later, sometimes as a problem to be solved.

**9.2. Review by: Gerry Leversha.**

*The Mathematical Gazette* **88** (511) (2004), 176-177.

http://www.maa.org/publications/maa-reviews/discourses-on-algebra

In this introduction to algebra, the author aims to show that the subject is no less beautiful, elegant and logically coherent than Euclidean geometry. The target audience in his native Russia is school students who need enrichment material in mathematics; in this country it is less obvious how such a text would fit into the national curriculum at A level, but it should appeal to able students and undergraduates who are looking for a more sustained and challenging course in elementary algebra. ... As the book progresses it becomes increasingly clear that the author has planned the development of ideas meticulously, not only so that he can constantly surprise the reader with the next turn of events, but also so that he can demonstrate unexpected links between results in algebra, number theory and probability. There are numerous ideas here for providing stimulating lessons for able pupils as well as for self-study, and the book would make a valuable addition to the school or department library.

**9.3. Review by: Mihaela Poplivher.**

*Mathematical Association of America*

Written by the leading Russian mathematician Igor Shafarevich, *Discourses on Algebra* is an advanced elementary algebra book intended to supplement the content of algebra courses in Russia, for students in grades 9-12 and their teachers. Its English translation can be used both for high school students and undergraduates interested in a deeper understanding of algebra topics, or in their preparation for mathematics competitions. ... Several very important facts presented in the book have several proofs. As an example, Chapter 4 on Prime Numbers contains many proofs of the fact that "the Number of Prime Numbers is Infinite", including Euclid's (most beautiful?) proof and Euler's proof. The chapter ends with the non-trivial Chebyshev Inequality for the number of prime numbers not exceeding *n*, included as a supplement at the end of the chapter. ... What I found particularly attractive about this book are the historical notes, the references to many mathematicians and their work, as well as the many original proofs included. In closing: I think that any student and any teacher interested in a deeper study of elementary (and maybe not so elementary) study of such topics as sets, polynomials, and numbers should read (pencil in hand!) this book. It may be particularly valuable for future teachers. The book is very well written, and it has detailed proofs and many exercises. Above all, this book will be remembered for its beauty and elegance.

**10. Linear algebra and geometry (with Alexey Olegovich Remizov) (Russian 2009, English 2013).**

**10.1. From the Preface.**

This book is the result of a series of lectures on linear algebra and the geometry of multidimensional spaces given in the 1950s through 1970s by Igor R. Shafarevich at the Faculty of Mechanics and Mathematics of Moscow State University. Notes for some of these lectures were preserved in the faculty library, and these were used in preparing this book. We have also included some topics that were discussed in student seminars at the time. All the material included in this book is the result of joint work of both authors.

**10.2. From the Publisher's description.**

This book on linear algebra and geometry is based on a course given by renowned academician I R Shafarevich at Moscow State University. The book begins with the theory of linear algebraic equations and the basic elements of matrix theory and continues with vector spaces, linear transformations, inner product spaces, and the theory of affine and projective spaces. The book also includes some subjects that are naturally related to linear algebra but are usually not covered in such courses: exterior algebras, non-Euclidean geometry, topological properties of projective spaces, theory of quadrics (in affine and projective spaces), decomposition of finite abelian groups, and finitely generated periodic modules (similar to Jordan normal forms of linear operators). Mathematical reasoning, theorems, and concepts are illustrated with numerous examples from various fields of mathematics, including differential equations and differential geometry, as well as from mechanics and physics.

**10.3. Review by: Erich Werner Ellers.**

*Mathematical Reviews*, MR2963561.

Igor R Shafarevich gave a series of lectures on linear algebra and geometry at the Moscow State University from the 1950s through the 1970s. Notes for these lecture are the basis for the present book by Shafarevich and Remizov. The book covers the basic topics that are usually taught in a course on linear algebra, including the Jordan normal form and quadratic and bilinear forms. In addition it deals with Euclidean, affine, and projective spaces as well as exterior algebras, quadrics, hyperbolic geometry, groups, rings, modules, and some representation theory. The authors stress the connection of linear algebra with other branches of mathematics, in particular with analysis and topology. In addition they frequently demonstrate the use of linear algebra in physics.

JOC/EFR January 2015

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