##
**Abstract algebra.
3rd ed.**
*(English)*
Zbl 1037.00003

Wiley International Edition. Chichester: Wiley (ISBN 0-471-45234-3/hbk). xii, 932 p. (2004).

This book under review is the third, again enlarged edition of one of the great recent standard texts on modern abstract algebra. The first edition appeared in 1991 with 658 pages (Zbl 0751.00001), while the second edition was published in 1999 with already 898 pages (Zbl 0943.00001), two additional chapters on commutative algebra and basic (affine) algebraic geometry, and various other enhancements and improvements.

The present third edition has been amplified by another 34 pages, mainly so by the addition of further currently important topics, namely Gröbner bases and their various applications, as well as by inserting additional exercises and streamlining parts of the original text. Generally speaking, this third edition appears as having been carefully revised, up-dated, enhanced, polished and even more completed.

As to the principal changes from the second edition, the authors have added a new section to Chapter 9 (entitled “Polynomial Rings”), which provides an introduction to the basic theory of Gröbner bases for ideals in polynomial rings, together with applications to classical elimination theory. Additional applications and examples have been woven into the treatment of affine algebraic geometry in Chapter 15, and a wealth of highly instructive exercises involving Gröbner bases, both computational and theoretical in nature, have been added to the text. Moreover, a number of new exercises have been added to other chapters, too, throughout the entire book, in particular to the special topics of general module theory, commutative algebra, and homology.

Now as before, this excellent textbook on modern abstract algebra is extremely comprehensive, versatile, didactically pronounced, mathematically rigorous, conceptually and methodically up-to-date, masterly composed and clearly written. The authors have carried out their guiding principle in writing this great textbook with utmost consistency and mastery, which was to demonstrate the power and beauty that accrues from a rich interplay between different areas of mathematics. The emphasis throughout has been to motivate the introduction and development of important algebraic concepts using as many concrete examples as possible. On the other hand, without trying to be too encyclopedic, the authors have set a high value on touching upon many of the central themes in modern algebra in a manner suggesting the very natural development of these fundamental ideas. A huge number of these important ideas and results appear in the exercises, whose total amount is matchlessly, almost incredibly high, and whose quality and bringing about are just perfect. In some instances, new material is even introduced first in the exercises so that students may obtain an easier access to it by a do-it-yourself approach, before getting led into systematic depth later on. In this regard, the book is really unique and outstanding, and there are other unique features that make this (meanwhile classic) text a true marvel, namely the unflagging lucidity and fullness, the cultured and vivid exposition, the great flexibility for students and instructors wishing to pursue a number of important areas of modern algebra, the authors’ hints for doing so, the rigorous and carefully thought-out proofs, the encouraging ease in explaining also the more advanced topics, and the wealth of material to choose from.

Certainly, the large scale of the book is unique, too, but this should not be considered a drawback or intimidation. On the contrary, this should be seen as one of the great advantages of the book, above all in view of its masterly combination of comprehensiveness flexibility, lucidity, theoretical depth, and richness of concrete examples. Indeed, this textbook of modern algebra offers all what it takes to become a steady companion of a student of algebra and mathematics, in general, and that from undergraduate studies up to advanced graduate studies. Also teachers and instructors, perhaps even researchers, will see it as an invaluable source of information, didactic guidance, examples and appropriate exercises.

As for the numerous exercises, which strongly vary with respect to their degree of difficulty, it would be highly desirable that the authors published a complementary booklet providing solutions of selected (hard) exercises. At least for average students, this could be of great help, further motivation and encouragement. However, and all together, the book under review is one of the greatest textbooks in modern mathematics, and the present third edition of it corroborates this judgement even more effectively. For the sake of completeness of this review, we briefly recall the contents of the book, which basically remained unchanged compared to the second edition (Zbl 0943.00001) from 1999, apart from the enhancements and improvements mentioned above. Part I is entitled “Group Theory” and contains the first six chapters. Part II, consisting of Chapters 7–9, introduces basic ring theory, including the new section on Gröbner bases.

Part III deals with linear algebra for modules and vector spaces, including tensor products as well as projective, injective, and flat modules. Part IV is devoted to the theory of fields and Galois theory, while Part V provides the fundamentals of commutative algebra, affine algebraic geometry, and homology theory, with particular emphasis on the cohomology of groups. Finally, Part VI discusses the representation theory of finite groups and contains the concluding Chapters 18 and 19. For the convenience of the reader, there are two appendices entitled “Cartesian Products and Zorn’s Lemma” and “Category Theory”.

As mentioned before, a lot of non-standard teaching material is contained in these six parts and nineteen chapters of the book, and the number of exercises, often with hints, after each section seems to be uncountable, just as the number of instructive examples scattered over the entire text.

Once again: This textbook is a pearl in the contemporary literature on modern abstract algebra!

The present third edition has been amplified by another 34 pages, mainly so by the addition of further currently important topics, namely Gröbner bases and their various applications, as well as by inserting additional exercises and streamlining parts of the original text. Generally speaking, this third edition appears as having been carefully revised, up-dated, enhanced, polished and even more completed.

As to the principal changes from the second edition, the authors have added a new section to Chapter 9 (entitled “Polynomial Rings”), which provides an introduction to the basic theory of Gröbner bases for ideals in polynomial rings, together with applications to classical elimination theory. Additional applications and examples have been woven into the treatment of affine algebraic geometry in Chapter 15, and a wealth of highly instructive exercises involving Gröbner bases, both computational and theoretical in nature, have been added to the text. Moreover, a number of new exercises have been added to other chapters, too, throughout the entire book, in particular to the special topics of general module theory, commutative algebra, and homology.

Now as before, this excellent textbook on modern abstract algebra is extremely comprehensive, versatile, didactically pronounced, mathematically rigorous, conceptually and methodically up-to-date, masterly composed and clearly written. The authors have carried out their guiding principle in writing this great textbook with utmost consistency and mastery, which was to demonstrate the power and beauty that accrues from a rich interplay between different areas of mathematics. The emphasis throughout has been to motivate the introduction and development of important algebraic concepts using as many concrete examples as possible. On the other hand, without trying to be too encyclopedic, the authors have set a high value on touching upon many of the central themes in modern algebra in a manner suggesting the very natural development of these fundamental ideas. A huge number of these important ideas and results appear in the exercises, whose total amount is matchlessly, almost incredibly high, and whose quality and bringing about are just perfect. In some instances, new material is even introduced first in the exercises so that students may obtain an easier access to it by a do-it-yourself approach, before getting led into systematic depth later on. In this regard, the book is really unique and outstanding, and there are other unique features that make this (meanwhile classic) text a true marvel, namely the unflagging lucidity and fullness, the cultured and vivid exposition, the great flexibility for students and instructors wishing to pursue a number of important areas of modern algebra, the authors’ hints for doing so, the rigorous and carefully thought-out proofs, the encouraging ease in explaining also the more advanced topics, and the wealth of material to choose from.

Certainly, the large scale of the book is unique, too, but this should not be considered a drawback or intimidation. On the contrary, this should be seen as one of the great advantages of the book, above all in view of its masterly combination of comprehensiveness flexibility, lucidity, theoretical depth, and richness of concrete examples. Indeed, this textbook of modern algebra offers all what it takes to become a steady companion of a student of algebra and mathematics, in general, and that from undergraduate studies up to advanced graduate studies. Also teachers and instructors, perhaps even researchers, will see it as an invaluable source of information, didactic guidance, examples and appropriate exercises.

As for the numerous exercises, which strongly vary with respect to their degree of difficulty, it would be highly desirable that the authors published a complementary booklet providing solutions of selected (hard) exercises. At least for average students, this could be of great help, further motivation and encouragement. However, and all together, the book under review is one of the greatest textbooks in modern mathematics, and the present third edition of it corroborates this judgement even more effectively. For the sake of completeness of this review, we briefly recall the contents of the book, which basically remained unchanged compared to the second edition (Zbl 0943.00001) from 1999, apart from the enhancements and improvements mentioned above. Part I is entitled “Group Theory” and contains the first six chapters. Part II, consisting of Chapters 7–9, introduces basic ring theory, including the new section on Gröbner bases.

Part III deals with linear algebra for modules and vector spaces, including tensor products as well as projective, injective, and flat modules. Part IV is devoted to the theory of fields and Galois theory, while Part V provides the fundamentals of commutative algebra, affine algebraic geometry, and homology theory, with particular emphasis on the cohomology of groups. Finally, Part VI discusses the representation theory of finite groups and contains the concluding Chapters 18 and 19. For the convenience of the reader, there are two appendices entitled “Cartesian Products and Zorn’s Lemma” and “Category Theory”.

As mentioned before, a lot of non-standard teaching material is contained in these six parts and nineteen chapters of the book, and the number of exercises, often with hints, after each section seems to be uncountable, just as the number of instructive examples scattered over the entire text.

Once again: This textbook is a pearl in the contemporary literature on modern abstract algebra!

Reviewer: Werner Kleinert (Berlin)

### MSC:

00A05 | Mathematics in general |

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |

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

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |

16-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras |

15-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra |