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I have a few weeks off from school soon, and I was hoping to self-study a bit of algebra. I don't think this question has been asked on here before, but does anyone have any suggestions for algebra books which are particularly suited to self-study?

Here is some background and motivation, if it's helpful.

I'm currently a junior in high school, but I have some familiarity with groups, rings, polynomials and fields, as I went through Fraleigh's book. I liked the writing style of that book, but I felt many of the problems were not helpful. I would like to get better at algebra with some other book, particularly one that's aimed at someone who already has a little familiarity with the objects, but is by no means completely comfortable with them.

One of my longer-term goals is to learn a bit of algebraic number theory, as right now I've only been studying elementary number theory. That's why I feel it would be good to learn more algebra first. (I'd also be interested in suggestions of other good subjects to learn prior to tackling algebraic number theory.)

At first I was considering using one of Lang's books, since they seem to be pretty universal, but I've heard rumors that they are very terse and not good for learning by oneself.

Any suggestions would be very appreciated, thank you!

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Flagging for community wiki. –  yunone Nov 24 '10 at 2:38
Dummit and Foote is a good one. –  user17762 Nov 24 '10 at 2:40
@Sivaram, thanks, I will check it out. –  yunone Nov 24 '10 at 2:45
I browsed around and found "A book of abstract algebra" by Charles c. Pinter. Based on the positive reviews, I decided to try it (it hasn't arrived, yet). You might find it useful, too. (Do note that there are two versions on amazon: there's one for about 110 dollars and one for about 11 dollars, but the content is, as far as I know, the same.) –  Max Muller Jan 6 '11 at 16:19

7 Answers 7

For Algebra you can look at these books:

  • Topics in Algebra by I.N. Herstein

  • Abstract Algebra by Dummit and Foote

  • Algebra by Michael Artin

  • Algebra by T.Hungerford (Springer)

  • Lectures in Abstract Algebra by N.Jacobson (Has 3 volumes!)

  • Algebra by Anthony Knapp. (2 Volumes.)

My feeling of Herstein is it has lot of problems which are challenging. For theory part i would like to use Dummit and Foote. Artin's Algebra is very well written and contains a lot of Linear Algebra. Anthony Knapps treatment of Algebra is very comprehensive, and contains a lot of Algebra. Since your aim is to read Algebraic Number Theory you might want to learn some Galois theory also for which there many good books like:

  • Lectures in Galois theory by Emil Artin

  • Field theory and its Classical problems by Charles Hadlock.

  • Galois theory by J.Rotman (Springer.)

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Thanks for the list of several choices, I'll be sure to browse through them. –  yunone Nov 24 '10 at 3:58
@yuone: No problem. Happy reading! –  anonymous Nov 24 '10 at 3:59
I first learned algebra from Herstein and still think it's a fabulous book for the strong student,despite being a bit old fashioned to today's algebracists. –  Mathemagician1234 Oct 8 '11 at 4:27
Addendum: If you choose Artin- make sure you get the second edition. It's MUCH better organized and clearer then the first and has many more exercises. –  Mathemagician1234 Jun 28 '12 at 3:53
Didn't Jacobson's books get rewritten into the two-volume Basic Algebra (which ain't that basic, i.e. it's also graduate level)? Is there a good reason to recommend the previous iteration of his textbook[s], i.e. the Lectures? –  Respawned Fluff Apr 11 at 19:58

I learned algebra (before taking Artin's course) by self-studying Nathan Jacobson's Basic Algebra I and II. This is a beautiful algebra textbook. However, it may help to supplement it with other presentations since it is a bit too concise in some places. A good choice for a supplement is Karlheinz Spindler's Abstract algebra with applications in two volumes. It has many interesting well-chosen motivational examples.

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Interesting, thanks for your suggestion Bill! –  yunone Nov 25 '10 at 7:50
+1 for Spindler, a very good and underrated book. –  Mathemagician1234 Oct 8 '11 at 4:25
Interesting book(s) Spindler's are. To groups he gets nearly at the end of the first volume; defined them being on p. 461! Motivation for groups being until then built mostly geometrically in the prior pages of the volume. (I could not help the Yoda act given the book's organization, sorry.) –  Respawned Fluff Apr 11 at 20:14

The absolute best book for self-study in algebra to me is E.B. Vinberg's A Course In Algebra. A Course In Algebra by E.B.Vinberg This book very rapidly became my favorite reference for algebra. Translated from the Russian by Alexander Retakh, this book by one of the world’s preeminent algebracists is one of the best written, most comprehensive sources for undergraduate/graduate algebra that currently exists. Vinberg, like Artin's more famous Algebra, takes a very geometric approach to algebra and emphasizes the connections between it and other areas of mathematics. But of special importance to a talented beginner and self study student like you, Vinberg‘s book begins at a much more elementary level and gradually builds to a very high level indeed. It also eventually considers many topics not covered in Artin or other books at the same "intermediate" level-including applications to physics such as the crystallographic groups and the role of Lie groups in differential geometry and mechanics!

The most amazing thing about this book is how it manages to teach students such an enormous amount of algebra-from basic polynomial and linear algebra through Galois theory, multilinear algebra and concluding with the elements of representation theory and Lie groups, with an enormous number of examples and exercises that cannot be readily found in most other sources. All of it is done incredibly gently despite the steadily increasing sophistication of the material. The book has a very “Russian” style-by which I mean the author does not hesitate to both prove theorems and give applications to both geometry and physics (!) throughout. Those who know me personally know this is a position I am very sympathetic to-and for there to be a major recent abstract algebra text that takes this tack is very exciting to me.

For anyone interested in writing a textbook on advanced mathematics, this is a terrific book to study for style. It is one of the most readable texts I have ever read. An absolutely first rate work that needs to be owned by any student learning algebra and any professor considering teaching it.And I completely and wholeheartedly recommend it to you as the best algebra book for you to begin with. If you master most of this book,not only will you find algebra immensely enjoyable and fascinating, you'll be ready for Lang or just about any other advanced text on the subject.

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The bad thing about Vinberg is that there are far too few exercises. –  user59083 May 31 '14 at 5:59

A Survey of Modern Algebra by Birkhoff and Maclane.

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Thanks Matt, I've heard of those authors before, so the book sounds promising. –  yunone Nov 24 '10 at 4:00
You have to learn from the best –  Sir Jective May 22 at 1:17

herstein's book is the best ever

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A bit of details? –  vonbrand Mar 28 '13 at 10:45

Higher Algebra by Hall and Knight

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Can you provide more details? –  Ludolila Nov 14 '13 at 19:07
This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post - you can always comment on your own posts, and once you have sufficient reputation you will be able to comment on any post. –  Dennis Gulko Nov 14 '13 at 19:30

You might like refer to previous MSE threads:

Requesting abstract algebra book recommendations


Good abstract algebra books for self study

After digesting them I just started with Vinberg's "A course in algebra"

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