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Any suggestions for a book on abstract algebra?

  1. Jan 11, 2008 #1
    Hello folks!

    Do you have any suggestions for a book on abstract algebra?

    Someone gave me this suggestion

    Algebra - Michael Artin


    however there are some bad (and convincing) reviews on amazon.com about this book (although the author is a MIT professor)

    and the price is to high...
  2. jcsd
  3. Jan 11, 2008 #2


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  4. Feb 14, 2008 #3
    Gallian's book is fairly standard (although I've heard it's a little easy), and an older edition might only be $20-30. I forgot the title (probably Abstract Algebra), but the author is Joseph Gallian.
  5. Feb 18, 2008 #4
    I liked Gallian's book ("Modern Abstract Algebra") and would recommend it.

    I referenced Herstein's "Topics in Algebra" at times during that course when I wanted clairity and I found it to be a great book as welll. I'm sure many hear will recommend it as well.

    Either can be found rather cheap and easily and won't lead you astray.
  6. Feb 21, 2008 #5
    Landin's An Introduction to Algebraic Structures is a nice little Dover book, could be good for self studying.
  7. Feb 22, 2008 #6
  8. Feb 22, 2008 #7
  9. Mar 4, 2008 #8
    I would suggest A Survey of Modern Algebra by Birkoff and Maclane a book which is most famous book on the subject and most tedious too, and another being Modern Algebra by van Der Waerden. These two books are enough to cover up the basics of abstract algebra but if your background is not sound you may find some part difficult, then rather study the a classic book in three volumes Lectures in Abstract Algebra Vol I,II,III by Jacobson
  10. Mar 10, 2008 #9
    I would say 'A First Course on Abstract Algebra 7e' by John Fraleigh is a pretty good book.
  11. Apr 3, 2008 #10
    I can't say my experience on algebra books is rich but when I stumbled upon ''Abstract Algebra'' by David S. Dummit and Richard M. Foote I was quite surprised. First of all it contains a large variety of topics. All the algebra that an undergraduate student of mathematics may need is in there. The most important thing about this book though is the way it is written. I remember myself actually enjoying algebra when first reading it. Apart from the fact I could completely understand what was going on, ''abstract'' seemed also reasonable.
    I should also mention that in my opinion this book is excellent for self-teaching.
    Here is the amazon link:
    and here you can find it as an ebook:
    Last edited: Apr 3, 2008
  12. May 24, 2008 #11


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    artin is the best. bad reviews of artin are from students who are not sharp enough, or not ready, or not willing to work hard enough to read it.

    it was for (sophomore) math majors at MIT, and obviously we are not all at that level.

    i would recommend it as a second encounter with abstract algebra ideally. before that, try my free online webnotes for math 4000, or ted shifrin's book, for which my notes are a companion.

    dummitt and foote is also quite readable and aimed at covering both undergraduate and beginning graduate algebra. so you can spend 2 years on it. the writing style is very clear, and there is an extensive and good problem set.

    i have a few beefs with it myself, which may not concern most students. e.g. when they treat diagonalization of matrices over the integers, they give an abstract proof in the text, and then when they expect you to really use it, they appeal to a more concrete version left to the problem sets.

    well i read the first negative review of artin on amazon and sure enough the guy practically said he was too lazy to work hard enough to read it. i.e. it wasn't dumbed down enough for him. if that is your mindset, he is right, it is slow reading, and things are not repeated or explained twice. this is not a bad thing however.

    and yes dummitt and foote is written in a more simple minded style. that is not nec a good thing however either. the reason artin is an undergrad book and df a grad book is that they cover more material. but artin is written for a stronger student.
    Last edited: May 24, 2008
  13. May 29, 2008 #12
    I also learned from Fraleigh many years ago and found it clear and to the point.
  14. May 29, 2008 #13
    You said the price is too high, but it's used for $30. If that's still too much then you're going to have to comb abebooks, alibris or go with a dover book.

    Artin was the book we used for the undergrad algebra class that I took. At the time I thought the book was dry, but then again at that time I enjoyed analysis ten times more than algebra! The cool thing I did remember was that Artin actually examines important groups, and ends up not falling into the trap of making it either linear algebra and/or number theory rehashed. The exercises are also excellent (but challenging).

    I'm currently studying Gallian, and he is very readable, there are many examples and the theory is well motivated. It makes for a very gentle introduction to algebra. His exercises are not nearly as challenging as Artin's and not as instructive. Also there is really not that much meat to the book, once you strip away the motivation, examples, history etc you are left with perhaps only twenty pages of theory. That makes it sound like I don't like it, but actually I do. It's a great first book on algebra.

    There was online algebra notes that I also liked, but since it was built upon learning by doing exercises, it might be too challenging to rec, I don't know. I have a dover book, but it doesn't compare to the other books on my shelf.
    Last edited by a moderator: May 3, 2017
  15. May 29, 2008 #14


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    here is another high quality book, comparable in quality to artin, in fact based on lectures by his father, and suggested above by algoman:

    Modern Algebra
    Van Der Waerden, B. L.
    Bookseller: A Squared Books (Don Dewhirst)
    (South Lyon, MI, U.S.A.)
    Bookseller Rating:
    Price: US$ 8.00
    [Convert Currency]
    Quantity: 1 Shipping within U.S.A.:
    US$ 3.99
    [Rates & Speeds]
    Book Description: New York Frederick Ungar Publishing Co 1937., 1937. Hardcover. Book Condition: Very Good. library bookplate on front pastedown; library labels on free front endpaper; 264 pages. Bookseller Inventory # 216909

    by the way, if you liked thomas calculus, you might like birkhoff and maclane, but it is really at a low level.
  16. Dec 10, 2008 #15
  17. Dec 10, 2008 #16


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    goodman's book looks pretty good, but i have a different approach. e.g. to prove thm. 1.9.18, he gives an explicit computation using inclusion - exclusion formulas and deduces the result that the phi function is multiplicative over relatively prime integers.

    to me it is much easier and more natural to prove that fact first and deduce the explicit formula.

    e.g. if n,m are relatively prime then it follows from the chinese remainder theorem that Z/nm is isomorphic to Z/n x Z/m. Hence their groups of units are also isomorphic. But a unit in the product is a pair of units, one from Z/n one form Z/m. hence the number of units in the product is the product of the number of units in the factors. QED.

    I.e. I like to use concepts to deduce formulas, whereas judging by this one example goodman uses computations.

    but the book looks worth a longer look. and its free. besides some people who are probably better teachers than i am have learned that concepts are hard for many beginners and computations are reassuring.

    welcome aboard professor goodman!
  18. Dec 11, 2008 #17
    Yes, that proof of the multiplicativity of the Euler function is given in Example 3.1.4.

    I'm no longer sure why I arranged things as I did, but apparently I wanted to introduce
    counting methods as part of algebra, which I could do in chapter 1 without building up any machinery.

    By the way, the proof of the multiplicativity of Euler's function using inclusion--exclusion is the one given by Dirichlet in his "Lectures on Number Theory" from 1863 (translated and republished by the American Mathematical Society 1999).

    Fred Goodman
  19. Dec 11, 2008 #18


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    professor goodman you re obviously a scholar of your subject and I could learn much from you. thank you for making this book available free.
  20. Dec 11, 2008 #19
    I'm planning on reading Topics in Algebra; I've read about half of Finite-Dimensional Vector Spaces, and I've skimmed most of Herstein's book, so I think I can finish the first three chapters over break (I'll probably go straight to field theory if I have extra time).

    I was thinking of buying Lang to supplement Herstein, for the category theory and commutative algebra (I'm doing some stuff with polynomial equations next semester), and maybe the homological algebra (I've been reading FAC, but I don't know how long that will last...).

    Is Bourbaki's algebra worth reading? Halmos recommends it at the end of FDVS, but I think the only other choices at that time were Van Der Waerden (also cited) and Birkhoff and MacLane (the earlier one).
  21. Dec 12, 2008 #20


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    skimming herstein's book is probably of almost no benefit whatsoever, working the problems on the other hand may help.
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