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Abstract Algebra (Was: Book recommendation!)


Main Question or Discussion Point


Next fall i will be taking Intro to Abstract Algebra so i was planning to give it a shot on my own during the summer break, but i don't know what would be a good book to buy online, that is not too expensive. I would like the book to be quite rigorous, like very proof based one, but that starts from pretty elementary stuff and that moves to quite abstract and difficult matters. I have looked some of them online, but i hve no clue what would be the best one. Based on your previous experience, what book would you suggest me to buy?

Thnx in advance!

Answers and Replies

I think I remember Mathwonk saying he had notes from his algebra class online. But maybe I made that up
Check the thread So You want to be a Mathematician there's about 100 pages of good info tot read and in that there are more books mentioned than you will ever be able to read
You are correct:


Scroll down for notes. I also advice reading the Linear Algebra notes as I'm becoming more and more aware of how important linear algebra really is.
What do you think of the following book? Has anyone worked with this book before?

Abstract Algebra
Third Edition

JOhn A. Beachy & William D. Blair

Is it acceptable for a first course in abs alg?

Science Advisor
Homework Helper
Gold Member
Just get Dummit and Foote. It has all the undergrad algebra you will need.

Also popular is the books of Artin and that of and Lang.
What do you think of the following book? Has anyone worked with this book before?

Abstract Algebra
Third Edition

JOhn A. Beachy & William D. Blair

Is it acceptable for a first course in abs alg?

That was one of the texts used for abstract algebra in my class. Its a very good option for a total beginner, goes over many elementary concepts about functions and such. I liked it very much though.
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Science Advisor
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That's the good thing with Dummit and Foote, it covers groups, to rings, to module (undergrad stuff) to homology, commutative algebra and representation theory (grad stuff)!
Re: Book recommendation!

sutupidmath> Next fall i will be taking Intro to Abstract Algebra... i don't know what would be a good book to buy online, that is not too expensive. I would like the book to be quite rigorous, like very proof based one, but that starts from pretty elementary stuff and that moves to quite abstract and difficult matters....i have no clue what would be the best one.

sutupidmath> What do you think of the following book? Has anyone worked with this book before? Abstract Algebra - Third Edition - Beachy amf Blair. Is it acceptable for a first course in abs alg?

quasar987> Just get Dummit and Foote. It has all the undergrad algebra you will need.
quasar987> Also popular is the books of Artin and that of and Lang.

ytoruno> Beachy and Blair - Its a very good option for a total beginner...I liked it very much though.

axeae> Isn't Dummit and Foote more of a graduate text?


First of all, to address Beachy - it is liked a lot and it is on a gentler level than some texts, like the many editions of Herstein.

If one were to use say:

a. I.N. Herstein, Topics in Algebra - Wiley 1975
I would say Beachy and Blair would be the ideal prerequisite and corequisite textbook for this. Herstein is a moderately difficult text, unusually clear. Most would say that Herstein is way too old and run to something like Hungerford's book. So yeah Beachy has a place in being a nice book before you break into the difficult medium difficulty abstract algebra books.

So in short Beachy and Blair's Abstract algebra is a great book...
others on the web [based on my notes for math texts] have stated:

'This excellent book was my textbook for 2 semesters of senior level abstract algebra.'
'this is the best mathematics book I own. I have used it as a suppliment while studying, in research, and in teaching. It is clear and readable.'
'the author takes a lot of time explaining proofs in the beginning. Over time, they leave more to the reader. The exercises are bountiful, and I often find a few interesting ones in each section. I highly recommend this text to anyone interested in higher mathematics. It's very thorough, yet very readable.'

I would say Lang's text(s) is/are good for really really advanced students, a third text or so is fine, otherwise he's abstract, dry and painful and wouldnt be appreciated as a first or second text. [see the quote under artin's textbook.

[Note: some of my quotes,are from my own personal notes, many are anonymous, a few people who's opinions i'e valued over the years are named, and a few references are unknown, the notes were merely for my own personal use, etc etc. Enjoy!]

Good Books
a. Algebra - Michael Artin

'This is my personal favorite textbook for teaching Abstract Algebra. It covers almost all of the topics needed for undergraduate abstract algebra and also reviews some linear algebra.' - Jason Williams
'Popular text, not my style'
'great algebra book, lots of different topics'
'If I were to pick one algebra book for undergrad algebra course, this would be it. Lang's undergrad algebra book may be comparable in the sophistication, but I find Artin more enjoyable.'
'the best book for bright undergrads' - Mathwonk

b. Abstract Algebra*- David S. Dummit and Foote

'liked by Alexander Shaumyan - New Haven, CT'
'This is a really good text in abstract algebra at the graduate level. - Fadi E'
'As a mostly self-taught graduate math student, I've come across a lot of really bad texts in math. It takes a while to hit on a really good one, but I think I've brought together a list of some of the best.'
'Compare this book to certain other ones (like Lang's Algebra, Hungerford's Algebra, etc.) and you'll agree, this one is way better. Most other books are too terse to study from, especially if you're studying on you own. But this one seems to cover the material pretty well, without falling into that trap. - Fadi E'
'standard first year grad student abstract algebra book'
'If this had been my first algebra book, I might have been intimidated. However, it was a great book for me to fill the gap between the undergrad and grad algebra courses.'
'I think it's an excellent undergraduate reference in that it has something to say, and often a lot to say, about precisely everything that an undergraduate would ever run into in an algebra class - and I'm not even exaggerating. I would say this is a good book to have on your shelf if you're an undergraduate because you can look up anything'
'overall I would use this book as a reference instead of a primary text, because the idea of reading it through from start to finish scares me.'
'I do not find the exposition readable and the fact that it was written by two authors is all to apparent. Every time I attempt to read this I want to burn it. However, there are a lot of problems'
'helpful for Qualifying Exams for Graduate School'
'Excellent for an introductory abstract algebra book; clear and comprehensive'
'This is a great book! It's introductory, appropriate for undergrads taking abstract algebra for the first time, but it is very comprehensive, useful for more advanced students as well.'
'Although it explains the material in great depth and at a slow pace, it does so in a logically sound manner. The authors provide rich motivation and always introduce material with an eye towards more advanced material to come later.'
'I find this book to be ideal for self-study and outstanding as a reference: it is very comprehensive and its presentations are clear. The book is still valuable to advanced students, both as review and for the advanced material'
'This book has a clearer presentation of some of the more advanced material than I have been able to find elsewhere'
'I would recommend this book for use as a textbook at all levels from undergraduate through graduate work, although for graduate students it may be better suited for use as a reference or for self-study.'

c. Algebra (Graduate Texts in Mathematics) (v. 73)*- Thomas W. Hungerford

[liked by Alexander Shaumyan - New Haven, CT]
[Grove and Hungerford liked way better than Jacobson]
[people think Hungerford is better]
[higher difficulty]
[much prefer this book for algebra, than Dummit]
[The bible.*Rigorous, with relatively straightforward problems.*Personally preferred to Lang's Algebra.]
[The field-theory chapter is horrible]
[The most enjoyable book on Algebra that I've ever encountered. There are parts of the proofs that are left to the reader, but I find that this keeps me engaged in the reading. Everything is done in full generality.]
[very comprehensive and well written]

d. Contemporary Abstract Algebra - Joseph Gallian - Houghton Mifflin

[Once you get abstract algebra and real analysis, you'll be ready for almost anything. There are three standard texts on abstract algebra, Fraleigh (easiest), Gallian (medium), and Artin (hardest, need your linear algebra).]
[Popular text, not my style, applied in flavor]
[This book is more than a find - it's a treasure chest. Gallian does more for the reader than just throwing a bunch of theory at them and laughing as the reader struggles to understand proofs and apply them himself; he provides a multitude of examples. And by a multitude, I don't mean three or four comprehensive examples per chapter: I mean three or four *per proof*. And often Gallian steps back a moment to discuss the proof or theorem he just outlaid in more conceptual terms, sometimes just as good as an example. The book is a delight, and absolutely perfect for the independent learner.]
[The most important feature of the book overall is its plethora of exercises and answers. Gallian averages about 45 exercises per chapter]
[I have read many texts but this is a the most beautiful and the most fun!]
[If you are looking for a rigorous step in abstract algebra this is probably not the book you want. If you are taking a fairly elementary one semester undergrad course and will never see this subject again, it is great. The proofs are weak (compare to Hungerford - the intro NOT the grad text - or Dummit and Foote - which, admittedly is more advanced, but not that much). This subject (like topology and real analysis) tends to depend on where you are and what you want.]
[This is the book I actually used as an undergrad. It was fun to read, and I remember getting really into it. It may be a nice alternative if you want something cuter than Artin or Lang.]
[Beachy and Blair is liked a lot on a gentler level - would be a good prerequisite and corequisite]

e. I.N. Herstein, Topics in Algebra - Second Edition - Wiley 1975 - 400 pages

[difficulty moderate]
[unusual clarity]
[While this book may have deserved 5 stars when it first came out, as it is very well written, it has disadvantages for students today...]
[a better introductory text than Artin according to some]
[Herstein's approach is to just concentrate on a few basic notions and then take it as far as possible before introducing new ideas]
[Given the present sorry state of American undergraduate education, it is likely that all but the strongest undergraduates will find this book a tough slog.]

f. Groups and Symmetry (Undergraduate Texts in Mathematics) (Hardcover) - M. A. Armstrong - Springer 1997

[For a first course in abstract algebra, this book is an excellent choice]
[This book is a gentle introductory text on group theory and its application to the measurement of symmetry. It covers most of the material that one might expect to see in an undergraduate course...]
[Excellent introduction to abstract algebra through group theory]
[This was the textbook for my first course in abstract algebra and the first "yellow book" that I read. I found it an excellent book: rather than starting with axioms and dryly deriving everything, it gets one to contemplate the meaning and motivation behind the axioms. This book will encourage you to play around with mathematics on paper and in your mind, helping you to get a concrete feel for a subject that many people view as painfully abstract.]

g. Abstract Algebra (Hardcover) - John A. Beachy and William D. Blair - Waveland Press Inc - 2 Sub edition - 1995 - 427 pages

[easier reading than Herstein, though both are great books]
[This volume offers a gentle introduction to proof in a concrete setting, the introduction of abstract concepts only after a careful study of important examples, and the gradual increase of the level of sophistication as one progresses through the book. It offers an extensive set of exercises that help to build proof writing skills. In addition, chapter introductions give motivation and historical context while tying the subject matter in with the broader picture.]
[Carefully develops proof writing skills]
[This excellent book was my textbook for 2 semesters of senior level abstract algebra. The unique feature of this book is that elementary number theory, equivalence relations, and permutations are carefully introduced at the beginning. Other books launch right into groups and then have to make long digressions to cover these topics. Comparing this book to the best-selling Contemporary Abstract Algebra by Joseph Gallian, I like that Gallian's book adds many applications which students will find interesting. However, Beachy and Blair's book puts a greater emphasis on developing student's ability to do proofs. The book also incorporates more number theory than many other texts. Answers to selected problems are included, so I recommend this book for self study as well as a textbook for any undergraduate abstract algebra course.]

h. Algebra - Serge Lang

[This book grows on you]
[Read Artin, then Dummett, then Lang]
[When I examined this book as an undergraduate I did not like it; often this is a sign that a book is poorly written, but in this case I just needed more background. Now I see this text as a gold-mine: clearly written, provocative, and rich in examples.]
[I recommend this book for any serious mathematician to add to their collection. However, it would be waste of time to read it until you already know a great deal of mathematics. This is one of those books that becomes a must-read once have already read 25 or so other serious math books.]
[a useful advanced graduate refernce on algebra - Mathwonk]

i. A Survey of Modern Algebra - Garrett Birkhoff and Saunders MacLane - [maybe AKP Classics]

[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.]

j. Modern Algebra and the Rise of Mathematical Structures - Second revised Edition - (Paperback) - Leo Corry - Birkhauser 2004 - 451 pages

[The book describes two stages in the historical development of the notion of mathematical structures: first, it traces its rise in the context of algebra from the mid-nineteenth century to its consolidation by 1930, and then it considers several attempts to formulate elaborate theories after 1930 aimed at elucidating, from a purely mathematical perspective, the precise meaning of this idea.]
[In the second rev. edition the author has eliminated misprints, revised the chapter on Richard Dedekind, and updated the bibliographical index.]


That's pretty much my 'wish list' and some of my 'recommended' books for other folks. Mostly through my struggles and searches in math and science, and for books that spoke to me, or seeing what texts old and new would be 'good texts or neat books for the shelf'.

Hope someone likes my notes, and recommendations *are* welcome,
Hhm. Well, a book that starts gently and you can cover quite a lot of ground in is the book by Fraleigh. I think it is a good book for what it's meant for, that is, for students just beginning with proofs. You can get the third or fourth edition cheaply.

If you want something more challenging, then try Herstein or Artin.
qspeechc> Hhm. Well, a book that starts gently and you can cover quite a lot of ground in is the book by Fraleigh. I think it is a good book for what it's meant for, that is, for students just beginning with proofs. You can get the third or fourth edition cheaply.

qspeechc> If you want something more challenging, then try Herstein or Artin.

Thanks for the comments. Fraleigh's been around for a while. My guess is lots of easy examples builds a solid foundation where it's most important... at the beginning. Older editions are always a great thing, sometimes older is better, sometimes newer is better, sometimes the editions are 95% the same, sometimes you buy on price, or by the cover, or condition. As long as you're happy with it, some books are awesome supplementary reading for $5-10-15 dollars, but might be totally questionable for $90. There are a few dover texts worthwhile too.

- Clark
- Warner
- Deskins

Clark isnt too easy but it's easier than most textbooks. Seth Warner's book has notation that's unique, covers an incredible deal, and ultra cheap, it's a linear algebra text and abstract algebra text in one 2 book set. There's Deskins book at 600 pages plus. And then the ancient 60s brutal modern algebra out of print dover books, great for culture, browsing, feeling lost, collecting....

Here's my notes [quotes] on Fraleigh:

37 A First Course in Abstract Algebra, 7th Edition - John B. Fraleigh
[not quite sold on this one]
[This book was my introduction to algebra, and I can say that with me it hit its target - I not only learned and understood abstract algebra, but I grew to love it and be thrilled by it. If you are outside of mathematics and looking for the way in, I don't think you can do much better than Fraleigh. You'll outgrow it - almost as soon as you put it down. But that's just testament to how far it can take you in just a dozen or so chapters.]
[I would recommend, if you can afford it, also buying a copy of a zippier book like Hungerford or Dummit & Foote (ask around) and using it together with Fraleigh. Fraleigh won't let you down in terms of giving you the space you sometimes need to grasp things (for example, he gives Tons of examples, and there are plenty of easy exercises that allow you to soak in patterns in the structures for yourself) and an advanced book will give you increased perspective and power.]
[Seventh Edition] - 2002 - 590 pages

I'd probably get Gallian and Armstrong and Artin as a core library. And then buy any easier or cheap used books that appeal to you. [like the Seth Warner book by Dover]
If one starts struggling with scary proofs at some point early on and you think you need more reading Fraleigh might help, one could run to Nicholson's Abstract Algebra text, Beachy as a more advanced abstract algebra text might help with a primer on proofs [some might get it with Analysis classes or texts to prep then for Rudin].

- Velleman's How to Prove It book might be good [for some] with an analysis text or with tackling abstract algebra, and some might take it before they get into advanced calculus.

- Eccles book on Mathematical reasoning is another,

- Cupillari's The Nuts and Bolts of Proof is one more,

- Rotman's Journey into Mathematics: An Introduction to Proofs [Dover].

Some folks might be strong enough from geometry or analysis or high end calculus texts with proof, some might tackle it with a Set Theory class and get hit with proofs.

- [Proofs and Fundamentals: A First Course in Abstract Mathematics*- Ethan D. Bloch]
- [Mathematical Proofs: A Transition to Advanced Mathematics*- Gary Chartrand]
- [How to Read and Do Proofs: An Introduction to Mathematical Thought Processes - Daniel Solow - Wiley 1982/1990]
- [Mathematical Reasoning: Writing and Proof - Second Edition - Ted Sundstrom]
- [Introduction to Mathematical Structures and Proofs - Corrected Edition - Larry J. Gerstein - Springer]
- [Thinking Mathematically - J. Mason]
- [Mathematical Thinking: Problem-Solving and Proofs - Second Edition - John P. D'Angelo and Douglas B. West - Prentice-Hall 1999]

A nice note i have on D'Angelo and West is:

[It covers proofs from all basic 'pieces' of mathematics and gives the reader a good feel for the 'proofology', both in technique and fundamental nomenclature and results, that a student is expected to know when taking the first analysis and 'abstract algebra' courses. It's not perfect though.]


[Difficult but well worth it]
[I'm using this in an undergraduate introduction to proofs class with a focus on analysis. As a freshman, it seems a bit overwhelming at times - I wouldn't recommend it to most freshmen or even sophomores. I do feel like this does a more than adequate job preparing me for more advanced math, and goes far above and beyond similar 'proofs and problem solving' style books.]
[The author gives solutions or hints for one-third to one half the problems depending on the chapter, which is more than enough for self-study. I would disregard the whiny one star review that is posted for this book; it is typical of someone who wants to be spoonfed mathematics.]
[For anyone interested in learning how to understand and write mathematical proofs, or a reference for college professors and high school teachers of mathematics.]


Sorry to make it a long post, but you got me thinking about *why* it's nice to have easier books on Abstract Algebra, and sometimes one needs to build a toolbox for 'proving' why some abstract generalized algebraic structure is 'so', and a few proof books might supplement nicely.

Knowing some of these books 'exist' can always help. We all know Polya's How to Solve it, Velleman's text when i first saw it gave me a sour comp-sci algorithmic heaviness, so i went on the hunt for more 'fun' books on the subject like D'Angelo and Eccles, and so on...

as someone with a brilliant comment, [my guess is one amazon review on How to Prove it] now anonymous in my 'notes' said about these types of proof books:

'When you have to take advanced calculus, topology or abstract algebra you will not be struggling to learn how to write proofs, you will be spending more time on learning concepts and little effort on the actual methods and techniques of proofs'

Learning the concepts is what you need, rather than struggling.

which is why it's nice to learn some subjects 'twice'
a. Take Calculus made Easy with Sylvanius P Thompson [easy]
then take it again with:
b. Courant's Differential and Integral Calculus, Blackie and Company/ or the 60s Fritz John/Courant rewrite.. [hard]

Heck, add the Frank Ayers Schaum's Outline on Calculus, JE Thompson's Calculus Made Simple [Feynman used it], Toss in Spivak, Leithold, Sherman Stein's 1972 Calculus text, and Apostol while you're at it]....

Take 3-4-5 times as long, but enjoy it, browse it, master it, know it deeply, and then sail onto whatever brings you joy and wisdom...

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