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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!

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  • $\begingroup$ If the book starts with an introduction to set theory, you don't need to be scared, in my opinion. $\endgroup$ – Git Gud Jul 25 '14 at 0:42
  • $\begingroup$ Basic Algebra I by Jacobson is a wonderful book. Effort reading it will be rewarded. $\endgroup$ – lhf Jul 25 '14 at 1:06
  • $\begingroup$ Well I'm glad, it's definitely worth the 11 dollars! Should I follow it up with Basic Algebra II, or is it years beyond me? $\endgroup$ – m1cky22 Jul 25 '14 at 1:16
  • $\begingroup$ Basic Algebra II is much harder, meant for graduate studies. $\endgroup$ – lhf Jul 25 '14 at 1:27
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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.

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  • $\begingroup$ I'll definitely check this one out! I haven't heard of this book yet, perhaps it's what I'm looking for. The problem with Dummit and Foote's book (which everyone recommends) is that it seems too fast paced for a beginner like me, but maybe after a few more courses I'll be able to grasp it a little better! $\endgroup$ – m1cky22 Jul 25 '14 at 1:06
  • $\begingroup$ I read DF later, and it's definitely an excellent book-- well-written, lots of exercises, and exhaustive. But it is definitely more advanced, and, while a degree of struggle is necessary to learn mathematics, too much difficulty at the beginning can be discouraging. Herstein will provide you with a good foundation, and help you to know what you want to do next -- be it DF or something else! $\endgroup$ – vociferous_rutabaga Jul 25 '14 at 1:19
  • $\begingroup$ Awesome! I just found a few new sources to study from! Thanks everyone! $\endgroup$ – m1cky22 Jul 25 '14 at 1:21
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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.

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  • $\begingroup$ I have started Dummit and Foote's Abstract Algebra, but after a few minutes I decided it was a little too fast paced. Maybe I'll give it another try, or I'll freshen up on some math I haven't done in a while and then try it again! $\endgroup$ – m1cky22 Jul 25 '14 at 1:05
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    $\begingroup$ Maybe you where the one who was fast paced when reading it $\endgroup$ – Jorge Fernández Hidalgo Jul 25 '14 at 1:38
  • $\begingroup$ That's hilarious, and sadly true. I looked through it again and it seems very reasonable. $\endgroup$ – m1cky22 Jul 25 '14 at 2:51
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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.

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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.

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  • $\begingroup$ Thanks! I'll definitely check it out! $\endgroup$ – m1cky22 Jul 25 '14 at 0:48
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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.

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  • $\begingroup$ I think this sounds like what I'm looking for as well! As for Dummit and Foote, as I said in other comments, I don't think I'm ready yet!! :( $\endgroup$ – m1cky22 Jul 25 '14 at 1:08
  • $\begingroup$ I'm actually reading the lecture notes right now, they are exactly what I was looking for! $\endgroup$ – m1cky22 Jul 25 '14 at 1:17

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