Introduction and Prerequisites to Abstract Algebra

Question

So I've seen similar questions asked, but none that really helped me out. I'm going to be a freshman in college next year, having already taken Multivariate Calculus and Elementary Linear Algebra. Of course these were very basic, only second-year undergraduate courses, but I am still interested in self-studying Abstract Algebra. I purchased Basic Algebra I by Jacobson, which is fairly readable so far (and I've been doing the exercises) but I am worried that I will get lost in it soon. Are there any better resources to start from, given my current mathematical standpoint? What else should I learn before digging into this topic on an introductory level, besides proof techniques and set theory? My problem is not in understanding the concepts, but rather my mathematical vocabulary is limited and I find certain definitions to be rather confusing.

Basically, where should I start/what should I read in order to build my mathematical vocabulary so that I don't constantly Google while reading texts on Abstract Algebra? Maybe if someone even has a math reference to symbols and words that would be great!

Answer 1

I'd recommend I.N. Herstein's "Abstract Algebra ". I read it when I was around your level, and found it very helpful and enjoyable to read. The proofs are elegant without being so terse as to make them difficult to parse; going through them helped me to develop a feel for proof-writing.

The first chapter introduces fundamental concepts. Some of these will probably be new and others will probably be good to review from more rigorous perspective: sets, functions, special types of functions, integers and their key properties, mathematical induction, and the basics of complex numbers. This is good material to be comfortable with, no matter where you head next in mathematics.

The rest of book focuses primarily on group theory, covering the fundamentals of the topic. After that, it includes a bit on theory of fields, abstract vector spaces, and polynomials, all important topics that you will see in greater depth later.

The exercises are particularly good because (i) there are many of them; (ii) they are grouped by difficulty -- make sure you understand all the easy ones, try at least a few of the more difficult ones; and (iii) they do a good job of introducing meaningful concepts, not simply providing busy work.

I might also mention that I was able to find a second edition of Herstein's book for a very good price, and the book is short enough for you to make some real progress on before the summer ends.

Answer 2

I would recommend Vinberg's book "A course in Algebra". I was making my own first steps in algebra using this book, and I think the book is great! It is not very fast, it's not too dry (it has a lot of nice examples and motivation), and it covers a big chunk of material. Another thing I really like in this book is the way it deals with linear algebra. There author is trying not to use coordinates, and I think it is very useful.

Answer 3

To get some background in the basic terminology of mathematics, you should check out Franco Vivaldi's excellent Introduction to Mathematical Writing, the lecture notes for a University of London course on mathematical writing. Of course, everyone at Math Stack Exchange welcomes questions about mathematical language! As for algebra textbooks, I cannot recommend Dummit and Foote's book highly enough. It has enough material for at least three fast-paced semesters of algebra, explains concepts clearly, and has a ton of good exercises.

Answer 4

Dummit and Foote's book Abstract Algebra is great, If you take it with patience you will find it is a book with great exercises and a good writing style, I definitely recommend it. You need some background to do the entirety of the exercises, but you can start it with only college algebra and some experience writing proofs.

Answer 5

Although I initially hated it, I would recommend (Michael) Artin's Algebra. It's different in that linear algebra is intimately tied into the flow of the book. But when I came back to understand linear algebra in terms of group theory, I found the book invaluable.

Abstract Algebra is slightly annoying in that several passes over the same subject in different directions is required to get a firm understanding. And in order to accomplish that, it helps to study abstract algebra from several different authors. I'd recommend stock-piling several books on it, just to come back to them later.