Algebra Seeking Recommendation on Abstract Algebra textbooks

[bacte2013]

Dear Physics Forum advisers,

My name is Phoenix, a sophomore with major in mathematics and an aspiring applied mathematician in the theoretical computing. I wrote this email to seek your recommendation on the textbooks for abstract algebra. I want to self-study the abstract algebra during this Summer to gain the knowledge of abstract algebra at undergraduate level, fall in love with the abstract algebra, prepare for upcoming undergraduate research in the theoretical computing, and (possibly) prepare for Abstract Algebra I course that I might take on Fall 2015 (the required text is textbook by Dummit/Foote).

Please let me inform you about my mathematical background: I took the computational single-variable calculus course and I am currently taking the computational vector calculus course. I self-studied the proof-writing book (Chartrand), therefore acquired the basics of proof methodology and set theory. I am currently self-studying the theoretical linear algebra (Friedberg/Insel/Spence) and mathematical analysis (Apostol, Pugh) textbooks. I learned the basic topics in linear algebra, such as determinants and matrix, through my Friedberg book and vector calculus books.

I bought I.N. Herstein's two books: "Abstract Algebra" and "Topics in Algebra" and borrowed C.C. Pinter's abstract algebra book because I heard that they are good books for beginner in abstract algebra. However, I often heard good things about M. Artin's "Algebra", and also books by Fraleigh, Gallian, MacLane, and Lang (undergraduate version); I particularly heard that Artin is the best algebra book for an undergraduate, providing both details and excellent insights; I also hard that Artin covers the linear algebra in the abstract level. Therefore, I am curious if I should purchase Artin and study it instead of Herstein and Pinter. I looked at some sample pages (Group chapter) of Artin and I seem to comprehend the presented materials but I am not sure if Artin will be a better book for beginners than Herstein and Pinter. I could use both Artin and Herstein/Pinter but I prefer to firmly stick with only one book to gain everything that author want to present in the book. Please provide me of your advice and experience regarding to abstract algebra books I mentioned and which book I can use for loving the abstract algebra! And please forgive me about this long post.

PK

 

[verty]

From the reviews I see of Herstein's AA book, I see the following:

"I cannot emphasize this enough: to get the most of this text you should do as many exercises as possible of a difficulty level that challenges you. If you do, you'll have a solid foundation of abstract algebra that will be more than sufficient if you choose to pursue graduate studies in math (or another field) later."

"There are HEAPS of exercises that vary in difficulty. Working these exercises is where the learning/understanding really happens!"

A common theme seems to be that the "Easy" questions are not easy, the "Middle" questions are punishing and the "Hard" ones are super difficult. But doing exercises is necessary because that is how one uses this book. And probably Herstein meant it to be that way because he does mention in the preface that it will be some readers' first exposure to abstract math. And he did write a second book so surely he meant readers of that book to have read this AA one first. So it can't be all that bad, really.

And it was published in 1999 which is not so long ago, and you do have them now, so I think I would forge ahead with those books.

Now for one of the negative complaints:

"Crucial results used to prove pivotal theorems are sometimes poached from exercises from earlier sections, so the book, damningly, is NOT self-contained. It is inexcusable to have the proof of Cauchy's theorem, for example, hinge on asinine parenthetical statements like "see Problem 31 of Section 4" or "See Problem 16 of Section 3, which you should be able to handle more easily now." What the hell is that about?"

So this is a possible worry. But what I have to say to this is, Herstein obviously wants you to have worked that other problem as a lemma, and why should that be a problem? If you think about it, referring to a lemma is what should happen, and having the lemma be a problem is also good if it is an important lemma, so that you get to understand the lemma. If the lemma is simply given, one may simply move on thinking it is easy and then miss the relevance of it when it is used.

Or more importantly, Herstein thinks of Cauchy's theorem as a consequence of that lemma. Do you not want to see it the way he sees it? So for these reasons, I don't rate this complaint very highly. Obviously it is a problem if one can't do the sub-problem. But then one can look it up online and will usually find a proof anyway.

I won't say anything about Artin except that it is newer, but 1999 is not so long ago.

 

[bacte2013]

^
Thanks for the reply! I am actually seeking more advice! Bump*3!

 

[WWGD]

I always recommend that, if/when possible, i.e., if/when you have access to a Math Library, that you go to the Algebra ( or other that you're interested in) section, browse through a few books and see which one feels right for you. You can look at the index, then you can try to see whether it has solved problems, etc. A great advantage of the more canonical (i.e., standard, most popular) books is that there are plenty of solved problems from these books. Of course, it is up to you not to become too lazy and use the solved problems to reinforce your work. I am a believer in wisely combining top-down with bottom-up approaches to learning.

 

[bacte2013]

^
I actually went to my university's mathematics library and read through D/F, Artin, and Herstein, and I actually enjoy them very much. What I am worried about is that which book will provide both insights and coverage as those two factors can only be understood by people who read them...

 

[bacte2013]

I am leaning toward Herstein and Artin as they are really good writers. However, I only want to stick to one.

 

[bacte2013]

I also just checked out the "Algebra" and "A Survey of Modern Algebra" by Birkhoff/MacLane, and they look really good! Should i stick with Artin and Herstein though?

 

[WWGD]

I don't really know what to tell you. Why don't you start with either one , see how it feels? Sorry, but the ultimate choice is yours.