Product Description
"Lang's Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books." - NOTICES OF THE AMS"The author has an impressive knack for presenting the important and interesting ideas of algebra in just the "right" way, and he never gets bogged down in the dry formalism which pervades some parts of algebra."- MATHEMATICAL REVIEWS

This book is intended as a basic text for a one-year course in algebra at the graduate level, or as a useful reference for mathematicians and professionals who use higher-level algebra. It successfully addresses the basic concepts of algebra. For the revised third edition, the author has added exercises and made numerous corrections to the text. ... Read more

Customer Reviews (24)

Terrible first algebra book
After having it used as my undergraduate algebra text, I hated algebra for about 15 years. Absolutely NO motivation or "genetic" understanding of the material. If you DO know the subject already, the book has many of the slickest proofs, so a good book for the professional, terrible book for instruction. Oh, another thing: if you want to know how to actually go about computing some algebraic object (eg, Jordan canonical form, Smith normal form, etc), do NOT try to find the answer in this book.

Comprehensive book, great for reference.
This book covers a very wide variety of topics in algebra, and has lots of exercises. It is a great book for reference.

Indispensable reference, but not perfect
Lang's Algebra is a very broad reference in abstract algebra: most subfields are at least mentioned, and there is a wealth of detail on many important topics. Overall, Lang's writing is clear and his exposition makes sense. The sheer quantity of information contained in this book, however, forces Lang into a less-than-logical general organization. His index makes up for that to a large extent, and this book is probably best used as a reference. Of course, Algebra makes an excellent reference and complement to any survey course in abstract algebra, because of its clarity and comprehensiveness.

That said, there are a few caveats that I would like to mention. Lang's terminology is nonstandard (eg "entire ring" for what other authors call an "integral domain"), and these accumulated idiosyncrasies become increasingly irritating as one reads through the book. Second, don't expect many applications. I found this to be a problem in Galois theory in particular, where Lang's exposition is completely clear and easy to understand, but seemingly pointless. He neglects even the classical and fairly abstract applications, like the Galois theorem, and he leaves infinite-dimensional Galois theory as an exercise. Now, there are not many applications in general in this book, so be aware that the clarity of Lang's exposition comes at a certain cost (ie, Lang never gets bogged down in too many examples or minor points). Third, Lang occasionally uses notation that he never defines and that does not appear in the index.

In summary, this is a wonderful general reference for algebra, but not a substitute for more specialized or more elementary textbooks in specific subfields, especially Galois theory. The best way to take advantage of Lang's clear, abstract exposition is to make a real effort to come up with examples on your own and use Lang for the general case. I wouldn't rely only on Lang's exercises either: they don't cover as much as you might want them to, but the ones that Lang does include are challenging and important.

Not a goodseller
There was no tracking for my book. I didn't know what happened. It took over 2 and half weeks to get it. I sent email to ask them about my book situation and got no answer.

Best Algebra Book.
This is among the finest references for Algebra I have ever seen.Comprehensive is an understatement.The only point of contention may be the terse language used throughout the text which makes it hard for novices.Great text to have on your shelf! ... Read more

Product Description
This is a self-contained text on abstract algebra for senior undergraduate and senior graduate students, whichgives complete and comprehensive coverage of the topics usually taught at this level. The book is divided into five parts.The first part contains fundamental information such as an informal introduction to sets, number systems, matrices, and determinants.The second part deals with groups. The third part treats rings and modules. The fourth part is concerned with field theory. Much of the material in parts II, III, and IV forms the core syllabus of a course in abstract algebra.The fifth part goes on to treat some additional topics not usually taught at the undergraduate level, such as the Wedderburn-Artin theorem for semisimple artinian rings, Noether-Lasker theorem, the Smith-Normal form over a PID, finitely generated modules over a PID and their applications to rational and Jordan canonical forms and the tensor products of modules.Throughout, complete proofs have been given for all theorems without glossing over significant details or leaving important theorems as exercises.In addition, the book contains many examples fully worked out and a variety of problems for practice and challenge.Solution to the odd-numbered problems are provided at the end of the book to encourage the student in problem solving.This new edition contains an introduction to categories and functors, a new chapter on tensor products and a discussion of the new (1993) approach to the celebrated Noether-Lasker theorem.In addition, there are over 150 new problems and examples. ... Read more

Customer Reviews (8)

Excellent.Don't get thrown off by "Basic" Abstract Algebra
This book will get you there if you believe in it.It has examples with solutions and problems with solutions.The only topic that does not have problems with solutions is categories.For this, I have the Hungerford text, and I am presently in the process of finding a better book for this.Otherwise it is the perfect book for self-study.

one of the best books on abstract algebra
I like this book very much. Pure math, all the proofs are complete and relatively easy to follow. Solutions for the odd numbered problems are provided. If you like math and want to learn the fundamentals of abstract algebra then this book is exactly what you need. It's written in theorem-proof/corollary style. I think every undergraduate student of mathematics, physics or information sciences should be able to use this book.

too concise in some parts, good elsewhere
I picked this book up at my students' society used bookstore for $10, it turned out to be a pretty good bargain. However, there are some theorems where the authors say something is obvious & I didn't think so. It isn't very often though, the rest of the book is pretty good, and I was a bit surprised because I had only heard of the well-known authors like Gallian, Herstein, Lang, etc. It covers maybe 3 courses worth of material too, including groups, rings, fields, vector spaces & modules, Galois Theory (complete with every possible application!), and more advanced stuff like a separate chapter on modules (in addition to the section with vector spaces), tensor products and principal ideal domains. There are also complete solutions to the odd-numbered problems. This book is surprisingly good except in certain parts, I like it.

A must for every math library
Bhattacharya is very concise and readable for a very difficult subject, if you are new to abstract algebra.His proofs are complete and expert and his outline is great.Also his problems are useful

Good but...
I agree this is a very clear and easily readable text but a graduate textit is not. One would need to know most of this material before startinggraduate level. Lots of stuff is left out and I think a rather better bookthat is equally clear and readable etc. but much more comprehensive withmore worked examples and historical motivation is Malik et.al. inMcGraw-Hill "Fundamentals of A.A.". ... Read more

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Customer Reviews (2)

One of the best intro to algebra texts
My first taste of algebra came from Lang's book (big mistake).I then read Gallian's undergrad. book and tried Lang again, it was still very difficult.I finally found Grove.This is a great book.Very easy to read and understand.Although it doesn't cover everything in Lang, it covers everything that commonly comes up in a first year graduate class.The book's only drawback is it's difficulty to find.If you're looking for a good intro. to algebra text (and if you can find it) buy it!

Well written and concise. Makes an excellent reference
This is one of my favorite beginning graduate texts. Although it follows a very terse definition-example-theorem-proof style, the proofs and examples seem very enlightening. Each Chapter comes
with numerous exercises interleaved with the material, as well as
exercises at the end of the chapter. Groups, rings, fields, and modules are covered in depth. ... Read more

Product Description
This text is aimed at the abstract or modern algebra course taken by junior and senior math majors and many secondary math education majors. A mid-level approach, this text features clear prose, an intuitive approach, and exercises organized around specific concepts. New to this edition are additional applications exercises to improve student learning. ... Read more

Customer Reviews (6)

Great book
Pinter writes a book the rest of us non-mathematical geniuses can understand.The book aims for the average higher level math student in group and field theory.I used it to supplement a college course in this area returning after 25 years out of college.The book is broken into nice, short chapters that progress well.A little light on isometries in R2 and a supplement with answers would be nice.

The earlier editions are just as valid and much cheaper.This stuff hasn't changed much in a century or two.

Clear, easy to follow, yet rigorous
This is perhaps the best textbook I've ever seen on any subject.It is extraordinarily easy to follow, yet very rigorous; it is an excellent self-contained course on group theory starting from an introduction to the topic all the way to Galois theory.The book is broken down into 33 bite-sized chapters with plenty of very good exercises at the end of each one (the only way to really learn a topic like this).In each chapter these exercises range from simple play-with-the-definition types to those that illustrate much deeper concepts.It is one of the few math books that even an imperfect student could probably successfully learn from very well, entirely on his/her own.

Not at these prices ($100+)
I am only giving it two stars because it is way over-priced. As a means of getting introduced to Abstract Algebra is really good. I also know that there are other sources that are also good. For one look at Groups and their Graphs by Israel Grossman. It is not as extensive as Pinter's, but will give you a great running start at this subject at a small fraction of the cost. Also try web resources, or your local university library.

Best introduction to abstract algebra
Pinter's book compares very favorably to other elementary treatments, such as Gallian or Birkhoff+Maclane. The author has intentionally deviated from the established modern writing style in mathematics texts (theorem...proof...theorem...proof...), presenting instead an eminently readable work of mathematical prose which can be understood by any conscientious ninth-grader. Pinter's straightforward proofs of "Cayley's theorem" and the so-called "Fundamental Homomorphism Theorem" alone justify the purchase.

This relaxed and readable style notwithstanding, a suitable level of rigor has been maintained throughout the text---which is fairly complete in its coverage of elementary topics. From basic group theory and ring theory to field extensions and Galois theory, many minor, auxiliary results are left as exercises to aid the student's facility in proof technique, while the important major results are explained in the most natural way possible.

Most importantly, Pinter's book would serve as an excellent second reference for those students approaching the daunting subject with another text. Once you've read Pinter, you can read just about any other text on the subject and immediately grasp. For this reason, it IS the best "first course" text, as previously stated. I consider it a masterpiece in its genre.

An excellent introduction to algebra.
The author does a good job of motivating the discussion, by describing how the next few pages relate to other areas of mathematics.The issue of motivation is extremely important; too many authors develop a very abstractview, and fail to properly motivate the student by providing an overview ofthe subject area. ... Read more

Product Description
This is a high level introduction to abstract algebra which is aimed at readers whose interests lie in mathematics and in the information and physical sciences. In addition to introducing the main concepts of modern algebra, the book contains numerous applications, which are intended to illustrate the concepts and to convince the reader of the utility and relevance of algebra today. In particular applications to Polya coloring theory, latin squares, Steiner systems and error correcting codes are described. Another feature of the book is that group theory and ring theory are carried further than is often done at this level. There is ample material here for a two semester course in abstract algebra.

The importance of proof is stressed and rigorous proofs of almost all results are given. But care has been taken to lead the reader through the proofs by gentle stages. There are nearly 400 problems, of varying degrees of difficulty, to test the reader’s skill and progress. The book should be suitable for students in the third or fourth year of study at a North American university or in the second or third year at a university in Europe. ... Read more

Customer Reviews (1)

Compact introduction to algebra
The first thing to not about this book is that it is dense.In my abstract algebra course we covered chapters 3-5 in an entire semester.However, it contained every proof we needed in class, and although it reads slowly, it was usually crystal-clear.Especially good is its treatment of group actions, which is sometimes lacking in introductory-level algebra books.There is a lot of additional material -- the author recommends covering chapters 1-7 in one semester, which may be feasible at institutions like MIT, but for the majority of students, the information contained in this book would be more than a year's worth.Chapter 8 contains almost a semester's worth of advanced linear algebra.Sometimes I wish there were a few more exercises in each chapter; especially in the beginning, because the proofs were relatively easy, it seemed like there wasn't a good enough selection of topics.The book is mercifully cheap, but even though it's paperback, it's very sturdy, and for such a low-cost choice, it has tons of content.Recommended. ... Read more

Product Description

A Concrete Approach to Abstract Algebra begins with a concrete and thorough examination of familiar objects like integers, rational numbers, real numbers, complex numbers, complex conjugation and polynomials, in this unique approach, the author builds upon these familar objects and then uses them to introduce and motivate advanced concepts in algebra in a manner that is easier to understand for most students. The text will be of particular interest to teachers and future teachers as it links abstract algebra to many topics wich arise in courses in algebra, geometry, trigonometry, precalculus and calculus.  The final four chapters present the more theoretical material needed for graduate study.

Product Description

Undergraduate Algebra is a text for the standard undergraduate algebra course. It concentrates on the basic structures and results of algebra, discussing groups, rings, modules, fields, polynomials, finite fields, Galois Theory, and other topics. The author has also included a chapter on groups of matrices which is unique in a book at this level. Throughout the book, the author strikes a balance between abstraction and concrete results, which enhance each other. Illustrative examples accompany the general theory. Numerous exercises range from the computational to the theoretical, complementing results from the main text.

Customer Reviews (4)

Somewhat terse but clear
Like all Lang books on algebra, everything is very clear.The book goes through group, ring, modules, field, vector spaces, Galois theory andsomeset theory including the Axiom of Choice.It does take some work and time to get everything, especially if you are getting acquainted with the topics for the first time.However, a motivated reader can certainly work through the book by himself.The only complaint I have is that the book often does not explain the motivation much.

Only for pure math majors
The book provides an introduction to abstract algebra for undergraduate students. It covers basic topics in the theory of groups, rings, fields, polynomials, vectors and modules.

The book is terse and succinct. It provides very few examples to illustrate the various definitions and theorems. The examples are sorely missing since it is probably the first encounter with abstract mathematical formalism for most students using this book.

The author does not provide motivations for most of the topics he presents. Many of the aforementioned topics are of use in physics, linear algebra and applied math, yet the author fails to mention the relevance of abstract algebra to these fields. Considering the fact that not all students using this book are pure math majors, this may leave many students asking "What is it good for?" too often.

The book contains many excellent exercises varying from the trivial to the highly challenging. However, no solutions are provided and no solutions manual is available. This makes the book highly unsuitable for self-study.

The major strength of the book is its rigor. The author covers many topics not covered by other textbooks and progresses very meticulously towards more complicated topics.He thus builds a strong foundation for future classes in abstract mathematics. For pure math students, the book is a valuable snapshot of how advanced pure math textbooks look like.

For other students looking for an introductory textbook on abstract algebra, "Abstract Algebra: An Introduction" by Thomas W. Hungerford is recommended. It provides a clearer presentation of the material and is better suitable for the beginning student.

Painful but satisfying
The title is quite misleading.It sounds innocuous but conceals some pretty neat stuff.This book is an introduction into modern algebra that starts with group theory and continues into Galois theory.It is terse at times but contains all the essentials.

This text should be required reading for all Computer Science majors who have an interest in cryptography or cryptanalysis.

nothing like his graduate text
I didn't enjoy this book at all. The author, as he states in the preface, has tried to strike a balance between rigour and brevity, while trying to set a standard for the American undergraduate algebracurriculum. The intentions are admirable; however, he doesn't quite pull it off.Most of the important topics are presented either in the text or in the problems. For that matter, they're well chosen, too. But quite often the presentation is either too brief, unclear, or devoid of motivation. Some definitions are ambiguous or imprecise, while the notation is sometimes clumsy. The only redeeming feature is the choice of problems. They vary in difficulty from section to section, but on an average are excellent(a factor that adds to their difficulty is that the material in the text is so skimpy). On the whole, I don't know what purpose this book serves. The average student is better off using Fraleigh or Gallian (or any of the host of easier books avalailable), while Herstein still has much to offer to the motivated one. Jacobson and Cohn(which I used a lot while taking my first Algebra course) are always available as outstanding references. ... Read more

Product Description
Using the proof of the non-trisectability of an arbitrary angle as a final goal, the author develops in an easy conversational style the basics of rings, fields, and vector spaces. Originally developed as a text for an introduction to algebra course for future high-school teachers at California State University, Northridge, the focus of this book is on exposition. It would serve extremely well as a focused, one-semester introduction to abstract algebra. ... Read more

Customer Reviews (2)

Rings, Fields, and Vector Spaces
This book is aimed to prospected High School teachers. The author's main area of study is Algebra. I would not recommend this book to someone looking for rigorous mathematics.
"Rings, Fields, and Vector Spaces" is a detailed book so much that I found myself lost while following proofs. This book is also outdated and is sold as quick-xeroxed copies at the bookstore of California State University Northridge for the price of about $12. I would recommend the newest edition along with "Introduction to Abstract Algebra" by Neal H. McCoy.

A must for allmath majors and aspiring math teachers!
I used this book for two classes and found it to be tremendously helpful.In both cases, it was a supplement to the required text. First, in my Foundations of Higher Mathematics course, I found it to be an excellent resource for understanding some of the basic concepts of divisibility in integers, gcds, primes and the Fundamental Theorem of Arithmetic. When first learning to understand and write proofs, one must have good understanding of the basic material. B. A. Sethuraman leaves no stone unturned when it comes to this. The text helps you learn how to learn mathematics, and this can easily translate into better proof writing.
Next, I found the book to be a valuable supplement to Algebra: Abstract and Concrete by Frederick M. Goodman, used in my Elementary Modern Algebra course. The explanations of rings and fields were excellent, but beyond this is a book replete with examples. When first discovering abstract algebra, one can never have too many examples!
This book is a beautiful combination of close attention to details, without being pedantic, as well as a broad overview of abstract algebra. I highly recommend it for all beginning math majors and aspiring math teachers!
... Read more

Product Description
This introduction to modern or abstract algebra addresses the conventional topics of groups, rings, and fields with symmetry as a unifying theme, while it introduces readers to the active practice of mathematics. Its accessible presentation is designed to teach users to think things through for themselves and change their view of mathematics from a system of rules and procedures, to an arena of inquiry. The volume provides plentiful exercises that give users the opportunity to participate and investigate algebraic and geometric ideas which are interesting, important, and worth thinking about.The volume addresses algebraic themes, basic theory of groups and products of groups, symmetries of polyhedra, actions of groups, rings, field extensions, and solvability and isometry groups.For those interested in a concrete presentation of abstract algebra. ... Read more

Customer Reviews (4)

Poor examples
Personally, I do not like this book at all.The idea is you teach yourself, which may appeal to some people, but not to me.Why should I teach myself when I'm paying tuition?The book gives you a few very obvious facts, then expects you to build everything up yourself.This ends up a) being a lot of work or b) making the exercises very difficult because you've missed all the middle ground.Not my style.

Hard to follow...
Personally, as an undergraduate, I do not like this book at all... It leaves a lot of important proofs to the reader which is not helpful when first learning Abstract.Plus definitions and concepts are not clearly states, a lot is "assumed."All in all, I would go for another book, or at least two supplements...

Good for problems and examples
We used this book in my standard undergraduate group theory course. I didn't particularily enjoy the writing style or flow of the book. So I didn't read it too much, but instead used my lecture notes to learn the material. However, the book does have a good collection of challenging problems. I also like how the book emphasizes geometry and geometric examples.

eh
We used it in my first course. I missed the book after it was stolen because it is easy and decently entertaining to read. I remember being bothered by the 'proof left as excercise' bits there are maybe too many for a first course...otherwise fairly content with it... This is a gentle start to real algebra. Afterwards, like post first semester, I would use something like lang herstein or van der waerden (if they still print it). ... Read more

Product Description
Now available in paperback, this classic, written by two young instructors who became giants in their field, has shaped the understanding of modern algebra for generations of mathematicians and remains a valuable reference and text for self study and college courses. ... Read more

Customer Reviews (4)

too concise for its own good
Any college level math text should have AT LEAST one of the following, but hopefully both:

1) plenty of examples illustrating the use of the taught proofs, theorems, algorithms, procedures, etc.

2) Answers to AT LEAST some of the exercises in the back of the text.

Without one of these, how would you ever know you were grasping the material?

Unfortunately, this book comes up pretty short on #1 and completely empty on #2. there are many sections that are lacking examples, and there are NO solutions to the exercises.

the book reads more like an ordered collection of theorems and corollaries. Not so great if you wanna teach yourself. Probably more ideal for decorating a professor's bookshelf.

A classic algebra text!Wonderful book...
This is a classic book on Algebra.There is much that I like about it.It is exactly what the name suggests--a survey course.It briefly introduces all sorts of topics, including rings, fields, groups, galois theory, vector spaces, lattices, boolean algebras, and much more.It is written at a fairly elementary level and it generally doesn't go into a great amount of depth in each subject.Interestingly, many (more modern) algebra texts omit a number of rather basic topics in this book.Also, many modern books separate "linear algebra" from "abstract algebra", whereas this book takes a more integrated approach.

I find it exceptionally clear and easy to read.Many of the subjects are made particularly easy; there is a strong concrete flavour to the text.The authors provide good motivation for the material.

I think this book would make excellent reading material for someone who is planning to study algebra.I did not pick it up until early in graduate school, and I wish I had had access to it earlier, when I was first studying ring and field theory.It is a fantastic reference for intermediate students, since it covers just about all the basics of algebra, and does so in a very understandable way.I think this book would make a fine textbook for an undergraduate course as well.

This is how algebra texts ought to be written
I have just started reading this book, and already I am
enthralled by the beauty and elegance of the authors'
exposition. Assuming nothing more than an acquaintance with
school algebra and a little geometry, they develop
the basic properties of central algebraic structures, including
rings, groups and fields. These are treated by reference to
familiar examples, such as the ring of integers and the
rational, real and complex fields. Everything that one learned
in school algebra is to be found here, though, as is to be
expected, each topic is treated at a rigorous, mathematically
sophisticated level. In the first two chapters, the properties
of the integers and rational numbers are gradually examined,
ultimately down to the definition of addition and multiplication
on the basis of Peano postulates. The authors then consider
polynomials, the real and complex numbers, vector spaces, linear
algebra and other topics.
The writing style is clear, concise and elegant, with each new
concept being carefully defined as it is introduced. The proofs
achieve a satisfying balance between detail and brevity. Indeed,
reading the proofs and completing the exercises would do much, I
am sure, to enhance a reader's mathematical facility.

If you are interested in acquiring a deeper understanding of
algebra, this book should serve as an excellent introduction.

A smorgousborg of symmetries of the square
Modern algebra is an extraordinary topic and Birkhoff and MacLane do a superb job of exploring it. However, as is often the case with mathematical texts, the material can be somewhat dry. ... Read more

Product Description

This is a new text for the Abstract Algebra course. The author has written this text with a unique, yet historical, approach: solvability by radicals. This approach depends on a fields-first organization. However, professors wishing to commence their course with group theory will find that the Table of Contents is highly flexible, and contains a generous amount of group coverage.

... Read more

Customer Reviews (2)

Good for future teachers
I believe this text of abstract algebra would work very well for future high school teachers in their training, since the book aims to answer a problem that is at the heart of high school algebra: solution of equations by radicals. Of course a high school student need not know Galois theory, but her math teacher should have some knowledge about the solvability and insolvability of equations. Also this follows the development of history, so all the "abstract" algebra don't seem too abstract with the motivation of problems and examples in history.

Don't use this book!
My professor wanted to use this book for an abstract algebra for students in a math education program. This book contained so many typos that it was difficult to read and understand the material the author was presenting. This book should have been revised in the last 3 years to fix all the typos. I feel this is both the author's and publisher's responsibility to the readers. ... Read more

Product Description
With the advent of computers that can handle symbolic manipulations, abstractalgebra can now be applied. In this book David Joyner, Richard Kreminski, and Joann Turiscointroduce a wide range of abstract algebra with relevant and interesting applications, fromerror-correcting codes to cryptography to the group theory of Rubik's cube. They cover basictopics such as the Euclidean algorithm, encryption, and permutations. Hamming codes andReed-Solomon codes used on today's CDs are also discussed. The authors present examples asdiverse as "Rotation," available on the Nokia 7160 cell phone, bell ringing, and the game ofNIM. In place of the standard treatment of group theory, which emphasizes the classification ofgroups, the authors highlight examples and computations. Cyclic groups, the general linear groupGL(n), and the symmetric groups are emphasized.

With its clear writing style and wealth of examples, Applied Abstract Algebra will bewelcomed by mathematicians, computer scientists, and students alike. Each chapter includesexercises in GAP (a free computer algebra system) and MAGMA (a noncommercial computeralgebra system), which are especially helpful in giving students a grasp of practical examples. ... Read more

Product Description
ELEMENTS OF MODERN ALGEBRA 7e, with its user-friendly format, provides you with the tools you need to get succeed in abstract algebra and develop mathematical maturity as a bridge to higher-level mathematics courses.. Strategy boxes give you guidance and explanations about techniques and enable you to become more proficient at constructing proofs. A summary of key words and phrases at the end of each chapter help you master the material. A reference section, symbolic marginal notes, an appendix, and numerous examples help you develop your problem solving skills. ... Read more

Customer Reviews (5)

Great condition.
The book was just like new as stated in the product description.It arrived before the estimated shipping date.
Thank you.

My Best Math Book
This is definitely one of my best math books. The main feature of this book is that the readers feel very easy to follow the provided concepts and understand what is important. I really don't know any other book as good as this one. Probably only David C. Lay's "Linear Algebra and Its Applications" can compete with this one. I truly recommend anyone who studies modern algebra to take a look at this. Good luck!

thats how math books should be written!!! (but plz, change that price there)
It is surely one of the books I most enjoyed!!
But its pricey, thats why a lot of colleges (or professors) try to avoid it.

The book has eight chapters:
1) Fundamentals
2) The integers
3) Groups
4) More on Groups
5) Rings, Integral domains, and fields
6) More on rings
7) Real and Complex Numbers
8) Ploynomials.

Definitions and Theorems stand out in Boxes, then later comes the examples!! (Plz Mathematicians who write books, just take a look here, see how nicely a book can be written, then go for the challenge).

one of the good things in this book, is that it does not assume you took a class in number theory before, so it introduces in the first two chapters everything (from a typical number theory class) that you would need in modern algebra class. (that might be a drawback for a student who took number theory class, and his professor is determined to start from the first chapter in this book).

other than the definitions and theorems stand out clearly, The author give examples on how that theorem can be used!! and The examples sometimes are really good!!

What's best in this book, are the problems after each chapter, they rank from direct applications to theorems, to CHallenging problems! (at least challenging for me). But note that some of the problems depend on each other! so if ur stuck on one problem, that means you might need to use a result from an earlier problem in the same chapter. its a drawback that the author does not say "use problem ... to solve this one", I think they assume that anyone solving the problems, is solving all of them in sequence, which what students SHOULD do. There is no way you can get a good grasp on the material in this book, unless you are a genius, or you solve ALL the problems after each chapter (at least a very good amount of them). I found best thing to do is try solving them in sequence, if you dont have time to solve all of them, then skip the ones that you REALLY think you can solve, and this way you can use the result later on.

I would recommend this book to anyone interested in modern (abstract) algebra! But I think a pre-requisite to self-study in this book is exposition to how to write proofs rigorously. (well sure thats the pre-requisite for any math course, but usually this subject is one of the first subjects studied in upper level math courses, and you better take another course that exposes you to how to write proofs, if your buying this book for self-study).

An excellent introduction to higher mathematics
I thoroughly enjoyed my modern algebra class, with an excellent professor and this excellent book. The book is very clearly written, and the concepts of sets, groups, rings, fields, and number systems are explained with detail. This is especially important since my summer research in number theory requires an understanding of these algebraic structures.

Great Book
I found the ideas in this book to be very accesible to the student with little mathematics experience (as I have).It is very straight foward, contains illuminating example problems, and even has an application sectionat the end of each chapter.Many abstract algebra books assume that youcan prove anything. However, Gilbert's book focuses on the techniques oflearning how to prove. ... Read more

Product Description

Prior to the nineteenth century, algebra meant the study of the solution of polynomial equations. By the twentieth century algebra came to encompass the study of abstract, axiomatic systems such as groups, rings, and fields. This presentation provides an account of the history of the basic concepts, results, and theories of abstract algebra.

The development of abstract algebra was propelled by the need for new tools to address certain classical problems that appeared unsolvable by classical means. A major theme of the approach in this book is to show how abstract algebra has arisen in attempts to solve some of these classical problems, providing context from which the reader may gain a deeper appreciation of the mathematics involved.

Key features:

* Begins with an overview of classical algebra

* Contains separate chapters on aspects of the development of groups, rings, and fields

* Examines the evolution of linear algebra as it relates to other elements of abstract algebra

* Highlights the lives and works of six notables: Cayley, Dedekind, Galois, Gauss, Hamilton, and especially the pioneering work of Emmy Noether

* Offers suggestions to instructors on ways of integrating the history of abstract algebra into their teaching

* Each chapter concludes with extensive references to the relevant literature

Mathematics instructors, algebraists, and historians of science will find the work a valuable reference. The book may also serve as a supplemental text for courses in abstract algebra or the history of mathematics.

Customer Reviews (1)

Shallow, bland and spineless
This book is basically useless. In a thoroughly unoriginal manner, Kleiner recounts the usual textbook story about the development of algebra, but with far too little depth and detail for anyone to gain anything but a smattering of clichés, and with no indication whatever of how this boring encyclopaedic catalog of who said what in what year is supposed to be "useful for teachers of relevant courses [and] for their students" (p. xi), which is the stated goal of the book.

The supposed usefulness of history lies in "showing how abstract algebra originated in, and sheds light on, the solution of 'concrete' problems" (p. 103). "History points to the sources of the subject ... It considers the context in which the originator of an idea was working in order to bring to the fore the 'burning problem' which he or she was trying to solve." (p. xii).

I could not agree more that history in this sense would be a wonderful resource for teaching and learning. Sadly, however, Kleiner's history is not of this type, his trumpeting notwithstanding.

Consider for example the origin of abstract group theory. "In 1854 Caley gave the first abstract definition of a finite group" (p. 31). According to Kleiner, this was "a remarkable accomplishment at this time in the evolution of group theory" (p. 31). How so? What was so "remarkable" about it? What was the "burning problem" that Cayley was trying to solve?

None of these questions are answered. Kleiner apparently thinks that no such justification is needed. Because Cayley's paper is agreeable to the modern mathematician it must automatically be "a remarkable accomplishment."

In conclusion, this book is less concerned with offering a meaningful complement to the establishment view of abstract algebra than with shamelessly and groundlessly touting the pedigree of the same. ... Read more

Product Description
Based on lectures given at Claremont McKenna College, this text constitutes a substantial, abstract introduction to linear algebra. The presentation emphasizes the structural elements over the computational - for example, by connecting matrices to linear transformations from the outset- and prepares the student for further study of abstract mathematics. Uniquely among linear algebra texts at this level, it introduces group theory early in the discussion, both as mathematical preparation for vector spaces and as a preliminary example of the rigorous development of informal axiomatic systems.

Topics covered include: sets and functions; groups and group homomorphisms; vector spaces and linear transformations; basis and dimension; matrices; representation of linear transformations; inner product spaces; determinants; eigenvectors and eigenvalues; and triangulation of endormorphisms and reduction to normal forms. A supplementary chapter discusses the connection between linear algebra and multivariate calculus and introduces categories and functors. ... Read more

Customer Reviews (5)

the ideal book for beginning undergraduate students
I fully disagree with the one star review...
This is a beautiful book though you have to belong to a certain reader segment to appreciate it.
The readers that will like this book probabely are beginning undergraduate students that want to build a mathematical career and want a first and quick introduction to abstract mathematics. The reader is not overwhelmed by exotic topics that are rarely used, but is introduced to abstract basic principles needed to understand other courses like for instance quantum mechanics,more advanced graduate courses in algebra or functional analysis.
The power of this book is that it covers just enough material to have a solid foundation of algebra for other abstract courses like functional analysis,
When I compare it for instance with the book of Shilov, I strongly prefer this book since it is better organised, covers less topics, but enough to know the basics. This book succeeds in providing shorter proofs compared to Shilov without sacrifying rigor and clarity. How is this possible ?? Ah my friend, this is a reward coming from abstract reasoning as illustrated by this book.

Is this really good?
I have just had a glimpse of all the three reviews here, and wonder whether the reviews are really true. I am now an undergraduate and taking a course which uses this textbook. Frankly speaking, I think this book sucks. If you carefully compare this book with Friedberg, Insel and Spencer's Linear Algebra, you will know the difference. This book misses out many things I consider important. Also, it does not contain many interesting and essential results, like all infinite-dimensional vector space has a basis. I don't think Zorn's lemma should not be mentioned in a book like this one.
Also, I can't really accept that it starts matrices like that. We all know that matrices come from linear transformations. Matrix multiplication comes from the composition of linear transformatiosn. This book does not mention this and start it right away. This approach is quite bad in my opinion.
Finally, I think that this book is messy. I think it is quite difficult to include introduction of linear algebra and abstract algebra at the same time, but if it's so difficult to do it, why don't you study it separately? I think the author tried hard to combine these two stuff, but failed to do that successfully. All my friends who have linear algebra and abstract algebra background agree with me. I definitely wont' recommend this book.

excellent
This book treats the basic principles of abstract algebra.
It is targeted to graduate students that need a more theoretical approach to mathematics (instead of the usual calculus courses)
This book is the best introduction to abstract algebra for the following reasons

-its style : good introduction in each chapter, making the reader curious to read further.
-its rigor : everything is well explained in full details with proof.
-its elegance : This book treats the abstract structural aspects of algebra and then suddenly shows how more concrecte applications follow from these abstract results. This is the kind of elegance and style that makes mathematics an art : build a very abstract theory and then see how more concrete stuff follows immediately as special case of this abstract framework. This way, new things can be discovered and most of the time (as in this book), you can explain practical calculation rules in a short and rigorous way.

Definitely the finest there is ....

math is art ...
Great book !
If you read this book, you will not only gain knowledge of abstract algebra, but also understand clearly why mathematics is art. It was a real fun reading this book. The topics are presented in such a way that the author leads you to a climax, making you curious to read further, and help you to explore the beauty in all the ideas of abstract mathematics. This book is the best book I ever read on abstract algebra.

The emphasis in on rigor and abstract structural concepts. It is nice to see that the more practical applications follow as a special result from the abstract structural concepts. This is a very elegant approach !!

A Great Structural Approach to Linear Algebra!
As a person who has a healthy interest in mathematics and has taken many classes, this is definatley one of the best!Professor Valenza taught it (he has been teaching this Linear Algebra class at CMC for ten years) and his book is essentially an excellent compilation of the lecture notes from his class.It takes a very different tack from most linear algebra texts: Usually, a linear algebra text begins by inroducing matrices and solving simultaneous equations, teaching computational methods.Prof. Valenza starts with the structure BEHIND all of that math however: Sets, Groups, and Vector Space properties.This structure is absolutely essential to knowing what's going on: My father took a (less superior) linear algebra class many years ago, and he never understood the concepts behind the mathematical manipulations; I actually sat down with him and taught him the things that I learned in Prof. Valenza's class.I really think that the knowledge in this book is invaluable to someone who wants to know what Linear Algebra is really about.

Just a few examples of the truly deep knowledge that this book communicates follows.For instance (this will ring a bell for those who have taken calculus) the "constant of integration" that must be added when doing an antiderivative is actually a property of group homomorphisms.The "absolute value" that must be introduced when taking square roots is structurally THE SAME property of group homomorphisms.Also, we all know that you can't divide by zero; it's just not allowed.But, the reason for that is ultimatley rooted in group theory; namely, the real numbers are NOT a group under multiplication.This type understanding has EVERYTHING to do with matrices and systems of equations!For instance, the fact that only square matrices can be inverted is a trivial consequence of a property of function mappings called "bijectivity." (a mapping from three- to two- dimensional space can't be bijective, for example)Many seemingly complex linear system problems can be simplified to a trivial questions by, for example, investigating the "span" of the column vectors of a matrix.There are countless problems that simply can't be understood withoutthe kind of structural knowledge that Prof. Valenza's book gives.

Understanding the basic properties that underlie so many mathematical objects has been a true delight for me, and anyone who wants to know what is really going on "behind the scenes" with linear equations would be wise to investigate Prof. Valenza's book.It's no accident that he also wrote a book on Fourier Analysis; understanding structure is simply the key to higher math. ... Read more

Product Description
Over the past decade, the undergraduate abstract algebraclassroom has undergone a dramatic transformation..Many faculty whowere exposed to new pedagogical techniques during the calculus reformmovement, wanted to experiment with those techniques in more advancedclasses.A variety of software packages were written or extended foruse in abstract algebra.This collection of article is the outgrowthof several gatherings of mathematicians who were interested indiscussing the teaching and learning of abstract algebra.

The Editors of "Innovations in Teaching Abstract Algebra" have chosen the articles appearing in the volume in order to acquaint readers about various technological innovations, to detail methods of teaching abstract algebra that engage students, and to share more general reflections on teaching an abstract algebra course. ... Read more

Customer Reviews (1)

Required reading for those who teach it
By the very name of the subject, the teaching of abstract algebra is something that one would think could not easily be done with a computer. That is false, and there are many existence proofs of that in this book. A collection of papers by instructors who have used new and innovative techniques to teach abstract algebra, these papers should be considered mandatory reading for everyone who teaches it.
When thinking about mathematics and the computer, your first thought is usually towards the major packages such as Mathematica and Maple. However, some instructors have developed other software and many of the papers in this collection describe their experiences. The software is easy to understand and in most cases is a free download. Students who walk in the door the first day of their abstract algebra class generally have many years of computer experience behind them. Not only do they expect computers to be a part of their education, they are reaching the point where they are beginning to demand it. When quality software is readily available and an instructor does not take advantage of such a powerful learning tool, they are doing the students and their profession a disservice.
Abstract algebra is a difficult topic for most people, and anything that eases the learning curve is to be embraced. From this book, you can learn enough to place it in a big, tight bear hug.

Published in Journal of Recreational Mathematics, reprinted with permission. ... Read more

Product 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. ... Read more

Customer Reviews (2)

Recreational but no-nonsense.
An off-hand account of algebra by one of the best authorities of thesubject. Recommended as a "serious" pass-time.

Modern Mathematics from the viewpoint of Algebra.
The author explains not just the definitions but also the 'philosophy' of algebra with highly non- trivial examples from geometry,analysis & topology not to mention algebra itself. Examples motivate the concepts and show their usefulness. There islittle number of formal proofs, but theorems areusually supported by natural 'explanations' whichmay be even more adequate. Frequently a single example in this book can replace a substantial theory of the corresponding subject. For example, about axioms of projective geometry. A reader with no knowledge of modern algebra cantry reading it ,but it would be more interestingif familiar with subjects of undergraduate mathe matics. Then an undergraduate reader will get some ideas how modern math uses sophiscated tools from algebra to understand other subjects. Another merit of this book is that it can besmoothly followed by the author's famous 2-volume textbook on algebraic geometry. ... Read more