Reviews (7)

Not the best book for undergrads....
This was the text used in both semesters of my undergrad algebra and I was really disappointed in it.The sequence we used was to start with rings and then into fields.During the first semester the instructor did an excellent job in making up for shortcomings in the text.The second semester (group theory) was a complete loss as I had both a bad text and a bad instructor.Joseph Rotman writes a FAR better algebra text, especially on the topic of group theory.I study algebraic topology and thank GOD everyday for Rotman!

A worthwhile pain in the....
This text was my first exposure to the beauty of Algebra and as my first text I must pay respect to Hungerford for his excellent, original and well written book.Hungerford has an uncany nack for presenting material in a straight-forward and consistent manner as well as providing a rich graded (i.e. they ascend in difficulty) section of exercises that, yes, do depend upon prior results.This dependence does not in any way limit the quality of the book since, such inter-connected-ness shows how certain seemingly un-related aspects are indeed related and, moreover, if you are using this text and have not noticed that this theme is prevalent throughout the book, then you may want to stop and take a closer look.Hungerford begins with the familiar integers, their basic number-theoretic properties and then uses these ideas, suitably abstracted, to introduce operations on and within rings all the while reminding the reader of the similarities.Only after an introduction to rings, their ideals and ring homomorphisms does Hungerford give the reader a glimpse of groups and their basic properties, again reminding the reader along the way how these operations are generalizations of the previous and more familiar operations.Now, the approach of Hungerford in this introductory text is definitely non-traditional since he introduces rings before groups and for some this may be a problem, why I am not sure, but it is pedagogically sound.Remember that in this day and age of American academia that most students have had very little exposure to rigorous mathematics and hence for the sake of most undergraduate students it is important to continually progress from the more familiar and less abstract (integers) to the less familiar and much more abstract (groups).Another positive aspect of this text is the inclusion of an appendix in which solutions and or hints to selected problems is contained, this feature is, again, beneficial to the student.As for those that require a student solutions manual, well my only comment to you is find another major that requires less work and or brain-power.Mathematics is about discovery, patience, persistence and truck-loads of hard work, which is partially realized as a direct result of struggling through difficult, challenging and often self-referential problems.Again, in defense of this book and the author, consider the following fact, Hungerford received his Ph. D under the direction of the legendary Saunders MacLane, so if you are at all familiar with the name then you should be familiar with his standards and hence should expect nothing less from the work of Hungerford.Thus, this book, aside from the ridiculous price, is a great introduction to abstract modern algebra.As for the negative side of this text, aside from what I have already mentioned, this book can be much too wordy and contains entirely too many examples for my tastes but these are petty and trivial.So what are you waiting for buy it (used).

Reviews (9)

Deceptively Wonderful
OK here's the truth:This book is an awful text when accompanied by not so great prof is teaching from it (e.g. one who delivers nothing but the text). BUT... once you begin to understand enough to know that the "trivial" "exercise" and "left to the reader" proofs are quite straightforward, the book is probably the best reference in Algebra you can hope for.

Well worth the read for any budding mathematician
I've been acquainted with several introductory graduate algebra
books over the years, and prefer this one for its coverage of all the fundamental areas (groups, modules, rings, linear algebra, fields, and category theory), being concise, and providing great care when outlining each proof.

If one compare's the amount of material in this book to Jacobson's "Basic Algebra Vol 1", Grove's "Algebra", or Herstein's "Abstract Algebra", Hungerford's book gets the nod.
Moreover, I much more prefer the concise definition, example, theorem, proof format over the more colloquial approach, as can be found in Jacobson's text. For me at least, the payoff for reading an algebra text is the beauty found in the logic and reasoning from which very profound results arise from the complex interaction and use of more straightforward ones. And this is exactly where Hungerford's book shines through in tremendous glory. When outlining a proof he does an outstanding job in citating the results from previous Chapters that are used. For me this is the strength of algebra (In geometry I cringe when I get a picture for proof, and in analysis it is often quite complicated to verify that a given situation possesses the appropriate conditions needed to invoke some famous lemma or theorem).

One last good word about this book: I found the exercises both in abundance (after each section) and quite reasonable for a first year grad. student. Happy reading.

The bee's knees
This book is the Basic Language of Mathematics (by J. J. Schaffer) of the Algebra world.Without doubt it is an excellent dictionary of general facts about algebra.But learning by it will leave one with at best amusing memories and a nervous twitch.Just for a taste, "This proof has two parts.The first is easy.The second is left to the reader."About half the proofs in the book go like this.And so at the end of each section, the reader is left with just the dry theorems to attempt the exercises, without the slightest idea of how problems of a certain type are actually proven or even approached.And oh, the exercises.A few are easy.A few are open problems.The rest in between seem to at one point have been at the core of someone's respective masters thesis.

This book has three genuinely good uses.If you have a doctorate in pure Mathematics, a respectable doctorate that has nothing to do with PDEs and the thesis for which took longer to write on paper then it did to format the pictures to fit the margins, and you want to look up how much of the ring structure of R is inherited by R[x] in under 3 minutes, then this book belongs on your shelf.

If you have taken at least two algebra courses at the graduate level (Real graduate, not graduate equivalent.Most of my Algebra I class had two pretty good undergrad algebra classes coming in, and got slaughtered by Hungerford), then this book can make for a good review of basic algebra you should already know.

Reviews (27)

The book after Herstein
I think I would only recommend this book to someone who has already had some exposure to algebra (or one especially gifted in mathematics). The beginning of the book is not too bad, but towards the end of Part I the pace quickens quite a bit. If you are willing to read over the text many times, and do all of the non-trivial exercises (there is an impressive olla podrida of algebra in them, most of which are the beginnings of some very deep ideas), then it should be a very rewarding experience. Namely because this is one of the most readable textbooks which covers everything from groups, rings, and fields to homological algebra and algebraic geometry. It is very rare to see this much material covered in one book, and for it to remain so structured (Rotman is an example of a book that covers a lot of material, but loses its structure somewhere).

Excellent book
I am surprised that this book has not got the 5 stars. It is very suitable for advanced undergraduates/first-year graduates. The book is full of examples; and the proofs are amazingly clear and succinct. The book introduces new concepts in the excercises long before the student encounters them in the sections.

This is a beautiful way to teach mathemtatics,--and indeed to learn it. The book is replete with examples that connect concepts from toplogy and real analysis with Algebra.

This book definitely deserves the 5 STARS.

Comprehensive book
I'm a graduate student in math. We used this book for the basic year-long abstract algebra sequence: group theory, chapters 1-4 and some of chapter 5; ring/field/galois theory chapters 7-9, 13-14. Some of my fellow students took a module theory course which was at least partially based off chapters 10 and (I think) 11. I'm sure more advanced courses could easily be based off chapters 15-end. Considering the cost of university books, I consider it very nice to buy one book for essentially 3+ courses.

The exercises in some sections are very diverse. My group theory professor made us do a huge number of them, and now I am amazed at how often I see questions similar to those from Dummit-Foote show up on past qualifier exams from many different universities. Regarding lack of answers in the back...well, you shouldn't need too many, and if you get really stuck, that's what the professor is for. And if you're learning it on your own then I'm thinking you should be brainy enough not to need answers!

The text itself is very readable and complete.

I don't think I'd recommend this as an undergrad textbook, although I've no doubt that there are some clever undergrads who could learn from it. I used Herstein's "Topics in Algebra" for my intro-to-abstract course as an undergrad. Herstein is designed to be introductory in nature, though still a wonderful book, while DF is more encyclopedic.

Reviews (4)

The worst mathematics book I have ever read!!!
I gave this book one star only because I couldn't give it a score of zero!!! Although many professors say that this book is excellent, remember they are professors who already understand the material. This book shows no examples, and the examples that it does show end abruptly with comments such as "all items are routine." Routine!!! Please show me what to do so that I don't have to spend more money on a separate study guide. Aren't mathematics texts expensive enough? This book may be an excellent addition to a professors library but this book should never, ever be used as a primary text for students.

Good for Self-Study
This is a tough book to review, because it is not clear who the real audience is supposed to be.The author says that it is aimed at first-year graduate students, with a bunch of extra material that can be referred back to during the second year and beyond.The earlier chapters also include efficient reviews (with sketched proofs) of material that should be familiar to those who have taken undergraduate algebra.

This characterization is debatable.Based on my experience reading most of the first six chapters (the first 400 out of about 1000 pages), I would say that the level of sophistication is roughly that of Dummit and Foote's "Abstract Algebra", which is usually considered an undergraduate book.D&F can sometimes be harder to read, and that is in part because Rotman's exposition is better (in my opinion), but also because D&F introduce more difficult material earlier.Whether D&F's approach is better is questionable; I find Rotman to be a much smoother read, but the organization is quite different -- for example, one does not encounter noncommutative rings until deep into the book, whereas Dummit and Foote introduce them immediately upon defining rings.On the other hand, early in the coverage of D&F's chapter on rings, one has to digest Zorn's Lemma and its applications almost from the beginning, whereas Rotman (I think wisely) pushes this back into a later section.In general, D&F introduce a lot of hairy examples that by themselves require a lot of effort to digest (thereby impeding the reader's progress through the core material), whereas Rotman's examples tend to be straightforward, at least as new concepts are being presented.

So, overall, the exposition flows more smoothly in Rotman's book, and the reader can cover the basics more quickly with less time spent on tangential examples and early generalizations.Also, Rotman's proofs are usually much cleaner and the overall style is very nice.It's more pleasant to read than Dummit and Foote.But this comes at a cost: Dummit and Foote do cover more material, and generalize at an earlier stage, than Rotman does.

But my biggest gripe concerns the exercises.Put simply, Rotman's are far too easy for what is being pitched as a graduate course.In fact, they are in general far easier than the homework problems I sweated through when I took honors undergraduate algebra. They're barely adequate to convince the reader that he has a basic grasp on the material, and there are almost no hard ones, let alone really tough, thought-provoking open-ended problems like one encounters in Herstein's "Topics in Algebra" (an undergraduate book).There are certainly no exercises in Rotman's book that would be of any use for a graduate student preparing for qualifying exams.They're not even much of a workout for a decent (honors student) undergraduate.

So, what is this book good for?I think it's great for reading material that is usually harder to understand elsewhere.Rotman has a real knack for clear mathematical exposition, and some of the chapters are a real joy to read.(Side note: there are also a lot of typos, at least in the first printing.The author maintains an errata list at his web site, and a second printing is coming soon.There are still many errata that he didn't catch, but they're fairly minor and do not detract significantly from the reading.) But this is simply not suitable for a primary graduate text or reference.Most good schools are going to demand more of their graduate students, and one is inevitably going to have to read Lang or Hungerford (and work through their exercises) to achieve competence at the graduate level.Rotman's book is a kinder, gentler book upon which to fall back when those books are inscrutable, as is all too common.I do recommend it highly for that purpose -- I think it's a very good secondary book.

An excellent Text
To begin with, don't let the title scare you.After having read through Rotman's book I am suprised that this text had not crossed my path earlier.It is a wonderful book and must have for any inspiring Algebraist.Moreover, I am quite shocked that the larger universities have not adopted this book.

(a)This book could quite easily be used as the standard third/fourth year undergraduate introduction to Abstract Algebra.In particular, the first four chapters provide a solid foundation for a moderate paced one semester course at which point the instructor has many different options for additional topics based on the performance of his/her class.

(b)Those students that move on to the graduate level, and obviously to a university using this book, would both be familiar with the temperment and flow of the author as well as devoid of the requirement of having to purchase another expensive Mathematics text.For example, my undergraduate Algebra text was Hungerford's and post completion the logical step, being familiar with his style, was to purchase Hungerford's graduate text.For those not familiar, let me tell you there is a night and day difference with repsect to how the material is presented.

(c) The remaining 7 chapters take the willing student on a pleasant tour of ring/module theory, some advanced group theory (for the inspiring group theorist I highly recommend the authors graduate text "Group Theory"), algebras(linear included), Homology(some cohomology) and finally some algebraic number theoretic concept under the heading of Commutative Rings III.

(d) Lastly, Rotamn does not get needlessly bogged down in any one section of the book.The flow is smooth, to the point with precise definitions, examples, and ample exercises.

Reviews (14)

good textbook. but....
During the lecture on the abstract algebra 1, Jacobson's book was the main textbook. Although his book is regarded as "bible" by some professors, I cannot help confessing thath i need another reference book which explains the material in more detail and more carefully. So I consult Frayleigh's algebra textbook whenever I hit upon something difficult to understand on jacobson's. Frayleigh's book is kinder, not to economize explanationsand examples. Jacobson is one of the top-class mathematian, but his fame doesn't assure his book's efficiency. His book is written too abstractively. I don't want to recommend this book to the beginers although it has the title "basic".

Excellent!
I'm truly shocked that this book has received such low reviews.This is not an easy book, but it shouldn't be faulted for that.It's clear and beautifully written, and it's been a pleasure to work through.Additionally, the chapters are divided into sections that are 'bite size' with exercises at the end of each, which has made it well suited for regular daily study.I would highly recommend it to any student with some mathematical maturity who wishes to get a good foundation in the subject.

Reviews (3)

Not that basic
It is indeed not an undergraduate level book, but Jaconson I&II
form a great overall introduction to algebra for the budding
algebraist. We used Jacobson I as the intro algebra text for
graduate students at Berkeley, and it can be recommended for
first or second year students (and beyond.) It forms a part of]
the trilogy of "Don't Let the Name Fool You" of not that basic
mathematics book with Weil's Basic Number Theory and Serre's
A Course in Arithmetic.

Excellent
Basic Algebra II is much better organized than Basic Algebra I, and it is an outstanding reference for the algebra that every Ph.D. graduate student should know.Perhaps the decision to prove Wedderburn's theorem via Morita theory was unfortunate since it makes the former seem more inaccessible than it really is, but otherwise I cannot fault the exposition. One word of warning: if your main interest is algebraic number theory, then Lang's _Algebra_ is probably a better reference since Jacobson omits several topics that are crucial for the number theorist.

Reviews (3)

A beautiful text
Fashions come and go, even in mathematics textbooks.Books from the 60s and 70s often have a set-theoretic flavor to them, because that was the fashion.Few books of any period are well written enough to resist such aging.spivak/calculus is one, and birkhoff&maclane is another.

This book strips algebra bare.I'm not talking about kiddie stuff like y=mx+c, I mean answers to the questions 'what does it mean to add two numbers together?' 'what is a real number'?Now there's no getting away from the use of sets and logic for these sorts of questions, but B&M do it with such elegance and clarity of exposition that it seems perfectly natural.And when you think about it, they're answering pretty fundamental questions; once that your school teachers glossed over.You can add 2 apples to 3 apples and count 5 apples, and maybe 2.5 and 3.5 apples make sense, but on what logical basis can you say that pi + pi is 6.28... given that you can never have exactly pi apples?Does saying you have a real number of anything make sense?

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.

Reviews (5)

Relatively readable book based on category theory
I won the Slovak translation of this book in a competition at the high school. It had many useful definitions and theorems in it and its rigorous, but readable approach seemed attractive to me and almost converted me into a rigorous mathematician, but obviously it turned out to be too abstract at the end.

However, the main problem was that several years later, because of particle-physics motivations, I tried to learn some things about the Lie groups, Lie algebras (e.g. exceptional Lie algebras), and their representations, and initially I assumed that such a big book had to contain something useful about these topics. It did not, I think.

This book is very good if you want to learn most of the rigorous math, partly based on the generalized abstract nonsense (also known as category theory), as it was known decades ago - the definitions and basic theorems about the basic algebraic structures. But you must be ready that many newest subjects - and those that are hot today - are not covered.

The book contains many problems - and these problems are solved in another book, which you may find very useful.

bold and beautiful
It has several sections not present in most introductory texts -- affine and projective geometry, multilinear algebra, and linear algebra (the latter only seen in Herstein's Topics in Algebra), category theory, and lattice theory. The first few chapters use permutations a lot for examples, later it uses matrix groups. We are talking about the 3rd edition here -- don't get an earlier edition!

THE algebra book, period.
After getting frustated by nearly all the so-called "authoritative" books on abstract algebra (Lang, Hungerford, Jacobson), I really can say that MacLane/Birkhoff is the best die-hard classic on algebra (I've already reviewed an out-of-print edition here at Amazon, but since that review is not reproduced in this edition's page, I'm doing it myself).

There's an interesting thing about the evolution of this book: the first edition has become famous among mathematicians, because it brought for the first time an elementary exposition of categories and universal constructions, directly from the horse's mouth (MacLane founded the theory of categories together with S. Eilenberg; Birkhoff was the creator of the theory of lattices), which is used as a basic tool throughout the book; it also contained unusual topics such as multilinear algebra and affine and projective spaces, but no Galois theory. The second edition has gained a chapter on Galois theory, but has lost the part on affine and projective spaces.

Reviews (4)

Good book but many errors
This is a good book, well-organized and contains many useful materials. However I do not recommend it as an independent study textbook (which is what I did). There are many errors, most of them are typographical but some of them are wrong numbers which are very difficult to correct, especially in the exercises given. And the exercises are very important -- the following discussions and proofs depend on some of them. Springer published a "corrected second printing" in 1999 but obviously they didn't do a good job (I purchased the new one and I got the old one in the library).

Fun but not easy
I am not sure I would take this specific book to a desert island but I would highly recommend it to anyone who is in possession of patience, self-motivation and willingness to spend time working through both the problems and any gaps that may occur within the proofs.This book can be used both by Graduate students or as a topics/independent study course for well prepared and motivated undergraduates.
In general, you can be guaranteed that the exposition present within any book by Rotman is par-excellence and this book is no exception.The material is presented in a sound and definite logical manner, which results in gently directing the reader through an 'introduction' to group theory by presenting the material in well thought out and bite sized portions.The only negative aspect of this book, which is minimal as best, is the authors choice of notation, for example the notation used to define G-sets in chapter 3 does not necessarily do the topic justice since G-sets in their own right are an important concept.But, as stated, this is the only complaint and is easily remedied.Enjoy!

Reviews (2)

A good book
This is a nice book that introduce the reader in a nicely way to Galois and Field theory. This book not only concentrate in the classic topics but also covers so much about areas not covered in other books. For example the topics of normality and separibility and pure separability are very well covered and the concepts of infinite extensions; among many others. This books also contains a nice feature and is tons of examples and exercises. Definitely a very good book. I took the course of Galois Theory with this text and I learned a lot.

Isbn: 0125998406
Sales Rank: 1380158
Subjects:  1. Algebra - Linear    2. Mathematics    3. Number Theory    4. Rings (Algebra)    5. Science/Mathematics    6. Mathematics / Algebra / Linear   

Isbn: 0387978372
Sales Rank: 431485
Subjects:  1. Algebra - Linear    2. Algebras, Linear    3. Linear Algebra    4. Mathematics    5. Science/Mathematics    6. Mathematics / Algebra / Linear   

Isbn: 0387944079
Sales Rank: 3207061
Subjects:  1. Algebra - General    2. Algebraic Fields    3. Galois theory    4. Mathematics    5. Polynomials    6. Science/Mathematics   

Reviews (2)

An undergraduate text for Mathematics students
This book covers the basics of finite field theory and its applications, but the coverage is at a level more appropriate for engineers than for mathematicians. Even for mathematicians, though, the books is an excellentintroduction to the subject.

Engineers may be frustrated by the lack ofhardware implementation issues.