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Does anyone have any suggestions for abstract algebra books 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 its writing style, but 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.

One of my longer-term goals is to learn a bit of algebraic number theory, as I've only been studying elementary number theory. That's why I feel it 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.)

I was considering using one of Lang's books, since they seem pretty universal, but I've heard rumors that they are very terse and not good for self-learning.

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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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  • $\begingroup$ Thanks for the list of several choices, I'll be sure to browse through them. $\endgroup$
    – yunone
    Nov 24, 2010 at 3:58
  • $\begingroup$ @yuone: No problem. Happy reading! $\endgroup$
    – anonymous
    Nov 24, 2010 at 3:59
  • $\begingroup$ 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. $\endgroup$
    – Mathemagician1234
    Oct 8, 2011 at 4:27
  • $\begingroup$ 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. $\endgroup$
    – Mathemagician1234
    Jun 28, 2012 at 3:53
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    $\begingroup$ 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? $\endgroup$
    – Fizz
    Apr 11, 2015 at 19:58
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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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  • $\begingroup$ The bad thing about Vinberg is that there are far too few exercises. $\endgroup$
    – user59083
    May 31, 2014 at 5:59
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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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  • $\begingroup$ Interesting, thanks for your suggestion Bill! $\endgroup$
    – yunone
    Nov 25, 2010 at 7:50
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    $\begingroup$ 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.) $\endgroup$
    – Fizz
    Apr 11, 2015 at 20:14
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A Survey of Modern Algebra by Birkhoff and Maclane.

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  • $\begingroup$ Thanks Matt, I've heard of those authors before, so the book sounds promising. $\endgroup$
    – yunone
    Nov 24, 2010 at 4:00
  • $\begingroup$ You have to learn from the best $\endgroup$
    – AnotherPerson
    May 22, 2015 at 1:17
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There are so many excellent books of abstract algebra, some of my favorite books

  1. Topics in algebra Book by Israel Nathan Herstein

  2. Abstract Algebra by Dummit and Foote

  3. Algebra by M. Artin

  4. A First Course in Abstract Algebra by J. Rotman

  5. Problems in Abstract Algebra by A. R. Wadsworth

I. Algebra A Graduate Course by M. Isaacs.

II. A History of Abstract Algebra by J. Gray

III. Abstract Algebra Applications to Galois Theory, Algebraic Geometry and Cryptography by Celine Carstensen,Benjamin Fine,Gerhard Rosenberger

Try one of the book from $1$ -$ 4$ and $5$ and if you have free time, you can study II or III to motivate yourself .

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