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I will be teaching a year-long undergraduate introduction to abstract algebra in the fall, and I am quite looking forward to it! I need to choose a textbook, and I don't have personal experience with any that I think will be suitable.

It seems that popular books are those written by Gallian, Fraleigh, and Beachy and Blair, among others. I can read the reviews on Amazon, and I can presumably also obtain copies of these and other books. (I have a copy of Gallian's, and it looks quite nice.) But browsing the books and the Amazon reviews gives me only a partial ability to understand what the differences will be in the long term from the students' perspective.

What are the substantial differences between these or other recommended books -- especially those which won't be immediately evident from reading their tables of contents or otherwise skimming the books?

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    $\begingroup$ Two favorites: Saracino's Abstract Algebra for most classes and Herstein's Topics in Algebra for honors classes with the best and brightest. $\endgroup$ – Gamma Function May 26 '14 at 1:11
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    $\begingroup$ @GammaFunction: I'm curious, why Herstein? I used that when I was an undergrad and it wasn't to my personal liking -- but I have heard the same sentiment expressed by others. $\endgroup$ – Frank Thorne May 26 '14 at 1:20
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    $\begingroup$ Seconding Saracino. $\endgroup$ – Benjamin Dickman May 26 '14 at 6:17
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    $\begingroup$ Please make the comments suggesting texts into answers, expanding a bit with overall discusson of strengths (and weaknessses) as you see them. $\endgroup$ – vonbrand May 26 '14 at 20:26
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    $\begingroup$ Please look closely at the prices. Some of the popular books have outrageous prices. Consider creating your own exercises so students don't have to get the latest edition (though then it could be hard to coordinate the reading assignments from multiple editions). I thought Beachy and Blair was fantastic for a second-semester course (on rings, polynomials, and Galois theory), but when I used it for a first-semester course I was shocked at how ill-suited it was for the kind of course I wanted to teach. Caveat emptor! $\endgroup$ – KCd May 27 '14 at 19:55
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An alternative approach is Childs' A Concrete Introduction to Higher Algebra (3rd ed., Springer 2008). It starts with some basic number theory, followed by rings and fields. Groups don't make an appearance until later. It's worth a look if you want to give your course a number theoretic flavor with applications and don't mind de-emphasizing groups somewhat.

(Personally, though, I quite like group theory and its many applications, so I don't know if I'd follow Childs' approach. That said, I'm happy to have his book on my bookshelf as a resource.)

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    $\begingroup$ There's not enough of a treatment of nonabelian groups to work well in an abstract algebra course for a whole year. I did use this text for several semesters when I taught a number theory course, for which I thought it worked quite well (I did not discuss continued fractions or the Moebius function, which weren't in that book, so it was fine). $\endgroup$ – KCd May 27 '14 at 19:53
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Judson's Abstract Algebra: Theory and Applications is different in that it is an open source textbook that is available at no cost. I haven't used it (yet), but I think it's worth pointing out for the aforementioned reasons. In addition to PDF and source versions, there's a web version that has proofs collapsed by default (handy for high-level reading and for students who want to try proving the theorems themselves first) and live SageMath cells.

See also the MAA review by Christopher Thron.

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  • $\begingroup$ I've used this and would recommend it. $\endgroup$ – Aeryk Oct 19 '16 at 14:23
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Perhaps implicit in this question is the issue of the content design of abstract /modern algebra courses at colleges and universities.

Some abstract algebra courses begin with groups, and go on to rings and fields. Some aspects of group theory are helpful in thinking about issues related to integral domains, rings, and fields.

On the other hand, for schools that train a lot of high school and middle school teachers, the integers are an example of an integral domain in the K-12 curriculum, so perhaps one should start an abstract algebra course with integral domains then go on to fields and groups? If one teaches something about modular arithmetic in K-12 perhaps having teachers who realize there is a concept such as a zero-divisor is useful and important.

http://en.wikipedia.org/wiki/Zero_divisor

The book one chooses may reflect the difference in approach to the content design of abstract algebra that is picked for the course at your school.

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I can speak to the books by Fraleigh and Beachy/Blair, since I've taken courses where each was the primary text. (Sadly, I got out of university teaching before I could teach undergrad algebra myself.)

I personally enjoyed Fraleigh's approach much more than Beachy/Blair's; however, as @mini mentioned above, a nontrivial portion of the teaching happens using problems. If I recall correctly, Fraleigh's recommendations to the instructor include spending the first third of every class session with students at the board presenting their solutions to problems. Later sections of the text also refer to results which were to have been proved in problem sets. So it may feel awkward to teach using this book if your classroom format differs significantly from the recommended format.

On other matters, I found Fraleigh's prose far more readable and clear than Beachy/Blair's, and I valued the fact that his definitions were more general -- e.g. not assuming that all rings have 1, which allows Fraleigh better parallels between subgroups/subrings and normal subgroups/ideals (if all rings have 1, then ideals aren't subrings).

Overall, Beachy/Blair is structured more as a journey to one particular big goal (Galois theory). Fraleigh is more of an exploration with some particular highlights. Since it's mostly number theorists who actually need/use Galois theory these days, I thought the "exploration" approach was preferable.

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    $\begingroup$ May not be accurate to claim that "it's mostly number theorists who need/use Galois theory these days". $\endgroup$ – paul garrett Aug 20 '14 at 15:53
  • $\begingroup$ Maybe you have a better sense than I do, @paulgarrett; what contemporary areas of active research are consumers of Galois theory? (Not more abstract Galois connections, I mean, but concrete Galois theory of the algebraic numbers.) I personally worked with arithmetic dynamicists wearing logician hats who used Galois theory, but I include them under number theorists for the purposes of this discussion. $\endgroup$ – MSmedberg Aug 21 '14 at 16:07
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    $\begingroup$ I'd claim that Galois theory is used throughout mathematics, often without fanfare, routinely, much as the ideas of "group" or "field" or "derivative" or "vector space" are. Apart from number fields, Galois theory describes the structure of field extensions and intermediate fields quite generally, and indispensably. The finite-field case is much used in crypto, and in error-correcting codes. Algebraic geometry very often uses Galois theory. There's a "differential Galois theory" applicable to differential equations... Galois theory is ubiquitous. $\endgroup$ – paul garrett Aug 21 '14 at 16:19
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My favorite is Abstract Algebra by I.H Sheth, printed in india!

If you do not have to be directly and immediately discuss on groups theory, this book will be useful. It covers more additional requirements comparing to Fraleigh's book.

IMO, the Fraleigh problem is its proof approach! They are not fast and clean as well as others. However it moves slow: step by step and will be a good choice for Intermediate students. though, it covers extra topics for BA.

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An unusual choice could be Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra by Cox, Little and O'Shea. It may not necessarily be a popular choice for a first course in abstract algebra, but I get the impression that it does get some mileage as an undergraduate introduction to aspects of algebraic geometry. The formal prerequisites are a course in linear algebra and a course involving doing proofs (in some situations, these "two" courses could be one and the same). Note that the book does not require prior knowledge of abstract algebra and the authors suggest that it could be used for a first course in the subject. Naturally, the emphasis is on fields and rings, rather than groups.

One potentially-attractive aspect of the book is the way in which it combines algebra, geometry and algorithms. There's plenty of material for a whole year, some nice applications to robotics and automatic geometric theorem proving, and an appendix containing suggested projects. You could also link up with other subjects such as geometric combinatorics (including polytopes) and algebraic statistics, although the latter could carry one too far afield for a first course. For the former, see Rekha Thomas' Lectures on Geometric Combinatorics.

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I've looked at Pinter's "A Book of Abstract Algebra" (Dover, 2010), and I like it a lot (haven't taught algebra, mind you, just a bit of use in Discrete Math classes). It is a bit slow, but gives plenty of concrete examples of application of the theory. Without that, abstract algebra will seem just mindless (and pointless) pushing meaningless symbols around.

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    $\begingroup$ It's a nice introductory book. A strong/weak point, depending on your point of view and students, is that it teaches much of the material using problems. $\endgroup$ – J W May 26 '14 at 20:19
  • $\begingroup$ @JW, as I said, introducing abstract algebra needs strong motivation, and that is easiest to provide by concrete applications, i.e., real-world problems. Not abstract enough if you are into the subject for it's own sake, sure. $\endgroup$ – vonbrand May 26 '14 at 20:24
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    $\begingroup$ Just to clarify, Pinter tends to introduce much of the material in the exercises. $\endgroup$ – J W May 27 '14 at 16:22
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    $\begingroup$ I think Pinter would work well as a supplement, but not as a main text. On the plus side, it's Dover and hence not costly. On the minus side, a lot of what I considered to be key material was in exercises or covered too briefly to serve as a main reading assignment. $\endgroup$ – KCd May 27 '14 at 19:52

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