Basic Notions of Algebra

Description

This book is wholeheartedly recommended to every student or user of mathematics. Although the author modestly describes his book as 'merely an attempt to talk about' algebra, he succeeds in writing an extremely original and highly informative essay on algebra and its place in modern mathematics and science. From the fields, commutative rings and groups studied in every university math course, through Lie groups and algebras to cohomology and category theory, the author shows how the origins of each algebraic concept can be related to attempts to model phenomena in physics or in other branches of mathematics. Comparable in style with Hermann Weyl's evergreen essay The Classical Groups, Shafarevich's new book is sure to become required reading for mathematicians, from beginners to experts.

by a Russian mathematician that Solzhenitsyn mentioned in his World Split Apart speech at Harvard in 1978; also the author of Russophobia(cf. Mark Weber's IHR article)

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.

Reviews

• "This is one of the few mathematical books, the reviewer has read from cover to cover ... The main merit is that nearly on every page you will find some unexpected insights..."

Zentralblatt für Mathematik und Ihre Grenzgebiete, 1991

• "...which I read like a novel and undoubtedly will become a classic. ... A merit of the book under review is that it contains several important articles from journals which are not all so easily accessible. ... Furthermore, at the end of the book, there are some Notes by the author which are indispensible for the necessary historical background information. ... This valuable book should be on the shelf of every algebraist and algebraic geometer. ("

Nieuw Archief voor Wiskunde, 1992)

• "... There are few proofs in full, but there is an exhilarating combination of sureness of foot and lightness of touch in the exposition ... which transports the reader effortlessly across the whole spectrum of algebra.... 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 modern 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..."

The Mathematical Gazette, 1991

• "... According to the preface, the book is addressed to students of mathematics in the first years of an undergraduate course, or theoretical physicists or mathematicians from outside algebra wanting to get an impression of the spirit of algebra and its place in mathematics. I think that this promise is fully justified. The beginner, the experts and also the interested scientist who had contact with algebraic notions - all will read this exceptional book with great pleasure and benefit."

Zeitschrift für Kristallographie, 1991

• "This is a truly wonderful book, one that anyone teaching algebra should read and which should be pointed out to talented students, particularly those who want to know a little more about what and why abstract algebra is. This book is volume 1 in the Algebra section of the Springer Encyclopedia of Mathematical Sciences … . The examples are particularly well chosen, simple enough to understand… . one that will enrich your understanding of algebra and deepen your knowledge of mathematics as a whole."

Fernando Q. Gouvêa, MathDL, March, 2007