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From what I understand, the book "A book of abstract algebra" by Charles C. Pinter is fairly elementary. On the other hand, the book "Algebra" by Serge Lang is very advanced. I'm assuming that just reading "A book of abstract algebra" is not enough to start Lang's book. From what I gather, the book by Dummit and Foote is somewhere in the middle - although probably closer to Lang's book. The problem is that the book by Dummit and Foote and the book by Lang overlap a lot and I can't afford both books. Now, I can maybe just read the book by Dummit and Foote, but I want to read Lang's book because it has more content and is more well known. So once I read "A book of abstract algebra" can I go immediately to Lang's book or is there some intermediate book I should read first?
I would strongly recommend reading the book by Dummit and Foote. It is a more elementary book than Lang's Algebra, but it is far easier to learn from. Lang's book contains a lot of material, but it is very dense, and it is better suited as a reference guide for someone that already knows the material than as an introduction for a new student.
It is very unfortunate that you find yourself with a budget constraint for texts and notes... As @peco says, perhaps you can find these sources (and many others) on-line, "for free" (at least "free" if you don't feel compelled to print them out). (E.g., I myself do put all my course notes online...)
It is true that the later editions of S. Lang's "Algebra" attempted to be encyclopedic... and even the earlier editions were impatient-terse (as was the man himself, who I knew a bit in the late 1970's). The Dummit-Foote book is somewhat less encyclopedic, but less "impatient", so, probably more readable. :)
I do recall my first encounter with an early edition of Lang's "Algebra", about 1972, after a previous diet of antiques. :) As a proponent of (what was then the somewhat novel) "Bourbaki viewpoint", it was a revelation... and really in a very good way. :) By this year, that novelty has largely worn off, but, still, I remember my amazement at the time... :) (Especially the brief mentions of categorical ideas, homological ideas, etc.)
A friend of mine, who is a high school math teacher and majored in math in college, recently asked me for a good book to read on Abstract Algebra (presumably, group theory). She is looking for something to read semi-casually, so no serious textbooks which hide intuition.
My own education largely skipped the basics. I read through Pinter in the library one afternoon, and that was pretty much it. So, can people recommend a good book on abstract algebra/groups for the casual learner?
A gentle introductory book full of intuition on group theory is Nathan Carter's Visual Group Theory. This answer is doubtless much too late for the OP, but may help others looking for casual books on abstract algebra.
Pinter is an excellent book, but I'd like to also recommend Abstract Algebra and Solution by Radicals, by John and Margaret Maxfield. It's very readable, and takes a semi-casual path through group theory, ending with elementary Galois theory and impossible constructions. This book was my first introduction to abstract algebra several years ago. If your friend wants a gentle introduction to the subject, I think it could be ideal.
The book introduces all of the major concepts in a way that is both easy to grasp and motivated. There are also interesting historical insights placed throughout the book. I had a friend lend it to me when I was starting to learn abstract algebra and it definitely helped pique my interest.
It depends the level of the casual reader (unless by casual you meant low prereqs). If you are looking for a book that requires a little more sophistication, but pays back two-fold in intuition I believe Artin's book Algebra is a good place to look. In particular, it helps motivate abstract groups by looking at the much more managable, understandable, notion of matrix groups.
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?
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.
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.
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.
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.
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; for the latter, see for instance Seth Sullivant's Algebraic Statistics.
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.
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.
Abstract Algebra is a branch of mathematics that studies algebraic structures such as groups, rings, fields, modules, and vector spaces. It deals with abstract mathematical objects and their relationships, rather than specific numbers or equations.
A textbook on Abstract Algebra is essential for anyone studying this subject. It provides a comprehensive and organized overview of the key concepts and theories, along with practice problems and examples to aid understanding.
When seeking a recommendation for an Abstract Algebra textbook, it is important to look for clear explanations, relevant examples, and a logical progression of topics. Additionally, a good textbook should have exercises and practice problems with solutions, as well as supplementary materials such as online resources or study guides.
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