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!
Best Answer
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.