Customer Reviews

Contemporary Abstract Algebra

5 star:		(5)
4 star:		(5)
3 star:		(1)
2 star:		(2)
1 star:		(0)

 

 

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.

I am a math major in my junior year, and this is the first textbook I have actually enjoyed reading. It is full of useful examples and it is clearly written and structured so that it is very easy to follow.

I'd recommend this book to any student who is looking for a great resource to help them learn/understand abstract algebra.

Don't let the math haters fool you. If anyone gave this book less than four stars it's because they were not ready for a proof based course or are otherwise incapable of reading or doing math proofs. I attempted this course three semesters ago and it was my first proofing course. I could follow in class but I could not do the proofs and I dropped the course. I took the course again three semesters later after a course on how to read and write proofs and after about five other proof based courses and I loved it! What a difference a little math maturity makes! My only disappointment is that we didn't finish the material on rings and didn't get to fields. I have one more semester before I graduate and this is going down as my favorite text and course. If you have some proof experience under your belt, you will do fine in this course with this text. If you've never had a proof course I wouldn't recommend this book. Gallian does expect the reader to fill in some of the details on their own.

After high school algebra and geometry, most of the math we learn doesn't apply much to the every day world. So, when a college math book takes the time to show examples of real world applications, I appreciate it. It helps to ground the material. This is not the rigorous, concise, law-theorem-corollary-lemma-and-repeat kind of book one gets used to, but it was fun to read. I'd recommend reading this along with a traditional style algebra textbook.

So far I have found the book easy to understand. The examples that are worked in the book allow me to easily understand the process and method to arrive at the correct conclusion.

Normally, mathematics textbooks are used as sleeping aids (Munkres' Topology, anyone?), but this one teaches you the concepts you need and doesn't do a terribly dry job of it. I'm assuming you're reading the review to learn the subject in independent study, because otherwise, who'd read a textbook review?

First of all, this book is thankfully small. Actually, quite a bit smaller than most mathematics textbooks I have been carrying around these years. But you didn't buy it for the size, so on to the contents and layout.

There are chapters and there are sections, like elements (latter) in sets (former); each section outlining some key concepts or theorems. The problems for each section do correlate closely to the course concepts. The author begins with a review of the foundations of mathematics and some property of sets and then begins with an introduction to groups before moving to a more detailed look.

With the curriculum in algebraic structures/abstract algebra fairly standardized, this book teaches you most, if not all, of what you would expect from a course in the upper undergraduate level. One thing you will not easily learn from the book, however, is that algebraic proofs rely more on a bag of clever, annoying tricks than some fundamental comprehension of the subject matter. And thus, if you ever find yourself stuck in a problem, ask your professor/mentor if you can borrow from her bag of tricks, or else you'd be kicking yourself in annoyance from unable to prove something so simple and elusive.

Overall, the book construction is fairly study, the material inside is comprehensive and fairly digestible, particularly with the subject matter broken down and explained as Gallian does. And, there you have it. Buy it, learn it, and then sell it/treasure it/burn it; you won't be looking at it again unless you hit mathematics graduate school.

I bought this book as a required text for a upper level undergraduate course in Abstract Algebra. I found the book to be a very good source for examples and problems. Many of the problems have solutions in the back which is a good feature. The book is thourough, rigorous, and at the same time easy to read. The ideas and concepts are presented so naturally that you almost forget you are reading a mathematics text.

The best intro ever, I have read many texts but this is a the most beautiful and the most fun! Trust me, buy it!

This book rocks! Galllian writes the book in language that is clear, precise,and understanable. A great reference book for any advanced algebra class!

Very general advice is very useless. I hope that the following comments are sufficiently specific so as to inspire a high school student to at least glance at the book and to help an undergraduate shy away from the book.

If you are studying pure mathematics for the first time, then I agree that a new take on exposition might be worthwhile. On the other hand, if you are serious about learning a subject, then a textbook which focuses almost exclusively on the subject is clearly the best choice.

To begin, the exposition in Gallian's book is tainted by the amount of unnecessary comments ranging from a short biography on a mathematician to superfluous quotes by Homer Simpson to insistent excerpts from Beatle's songs. If you want to study math history, then buy a book on math history; if you want Simpsons, look up some episodes, etc. etc. A valid question is: "But what if I want to know about a topic's history and applications?"...Answer: go to Wikipedia or ask a professor.

Gallian's confused exposition can be made slightly clearer by appropriately identifying its audience:

High school students: This text serves as a very good, but not excellent, introduction to abstract algebra. The first few pages of the book are quite decent for an introduction to elementary number theory with proofs. My advice would be to skip the "extra" sections on things like error correction codes in preference of focusing on learning the mathematics. The exercises in the first chapter, as they are throughout the entire book, are not exactly what a mature student would call "exercises", but instead "drills". Doing such drills will help you absorb the material, especially if you are studying this book on your own. You will learn the basics of cyclic groups and rings.

Undergraduate Math majors: Assuming you know a little bit about linear algebra, say at the level of Lang's "Introduction to Linear Algebra" or some introductory real analysis, then optimize your time by avoiding this book and instead study something serious, such as M. Artin's "Algebra" or Dummit and Foote's "Abstract Algebra". The exposition in this book is so clouded and the exercises so routine, that there is almost no way this book will prepare your for graduate studies in mathematics. On the other hand, Artin's "Algebra" D&F's "Abstract Algebra" will most certainly do the job. Compare, for example, Artin's section on equivalence relations and cosets to that of Gallian's. Artin introduces these concepts early on, which makes fundamentals like Lagrange's theorem and the first isomorphism theorem useful and natural. Gallian instead decides to delay these concepts until after he has defined isomorphisms (even more tasteless, homorphisms come after isomorhpisms). In addition, compare D&F's section on ring theory to that of Gallian's. D&F are succinct about the basic axioms (a0 = 0, etc.) and get right to structures like division rings to yield illuminating concepts. Gallian instead dedicates a whole chapter to almost useless things like "subring tests" (you don't need "tests" if you know the definition). These are only but a few of the drastic differences in exposition you will find. Finally, and perhaps not as important, is the notation. It is poor taste to use "Zp" to mean the ring of integers modulo p, versus what grown-ups label as "Z/pZ". In addition, the notation for cardinality could have been better: Gallian uses |G| instead of the unambiguous #G ( the |.| notation should be reserved for absolute value).

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Here is another telling part of the book: in proving Thm. 20.1 (pg. 352) that given a field F, an irreducible p(X) in F[X], then there exists an extension of F such that p(X) has a root: our author clutters the main point with useless commentary and four strings of equations. If you know what "projection" and "isomorphism" mean, then this "proof" is really just a one-liner, as follows: If x is the projection of X, where x is in F[X]/(p(X)), then p(x) = 0. done.