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Abstract algebra

  1. Aug 11, 2011 #1
    Currently I am reviewing basic algebra, trigonometry and I will also be starting calculus this fall semester...
    I enjoy reading about math and I wanted to know what abstract algebra is? Would this be to difficult to read seeing that I am only starting calculus?

    If so what other types of books could I read that are advanced and at the same time I could understand for my level of mathematics.

    Thank you.
  2. jcsd
  3. Aug 11, 2011 #2


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    Abstract algebra is the study of algebraic structures: groups, rings, vector spaces, etc. It's a very important branch of mathematics and provides the foundation for a lot of well-known results.

    Yes. There are a couple of reasons why:

    1) You probably haven't been exposed to that many proofs if you're just starting calculus. If this is so, it will be very difficult to absorb the material. Abstract algebra is laden with theory and if you aren't accustomed to writing proofs, you can read something and nod your head and think you're understanding it, but when it comes to proving results, you'll have more of a difficult time.

    2) Linear Algebra is a part of Abstract Algebra. It's common to have taken linear algebra to get a feel for vector spaces and other algebraic structures before delving into groups, rings, modules, etc. While linear algebra certainly isn't a prerequisite for abstract, this class definitely is a plus to have before taking it.

    There are lots of books out there. The basic stuff probably won't be too bad to retain. However, the more advanced topics will probably look like a foreign language.


    On a side note, I don't want you to get the wrong impression and think that you shouldn't look at new material. It's great to have an inquisitive mind, but just realize that without the mathematical maturity and background that it will be more difficult to retain the material.
  4. Aug 11, 2011 #3
    Thank you for the reply.. can you recommend a field that I can study... such as linear algebra? or is that to advanced to.?
  5. Aug 11, 2011 #4


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    I would suggest looking into a proofs book. You'll be way ahead of the game if you invest time in learning how to write proofs. After this, you'll be much better equipped to look at linear algebra and abstract algebra. Also, when you do end up taking proofs, it will be second nature to you by then.
  6. Aug 11, 2011 #5
    I don't quite agree here. I think reading a proof book can get quite boring and can be quite unmotivating. In my opinion, it is best to learn proofs while applying it to something. And I feel that abstract/linear algebra is the perfect place to learn proofs.

    Pick up a good but easy abstract algebra book like Pinter's "a book on abstract algebra" and work through it. Make ALL the exercises. Read a proof book concurrently to understand the little annoying details about proofs. This way you'll learn abstract algebra and proofs!!

    Linear algebra is a nice topic to learn as well. Maybe linear algebra is better because it doesn't involve proofs right away. Pick up Schaums outline and work through the exercises. After you read the book, you'll know a bit what linear algebra is all about. You might want to read a more difficult book after that, like Friedberg or Hoffman & Kunze.

    Another great place to learn proofs is while learning discrete mathematics (like combinatorics and graph theory). The book "discrete and combinatorial mathematics" by Grimaldi is an excellent book for this!!
  7. Aug 11, 2011 #6


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    As far as motivation goes, I agree. It's very easy to get lost in trying to learn proofs and wondering why you're doing it.

    I agree for the most part, but for someone trying to learn both proofs and abstract algebra (without the aid of a classroom setting), you have to agree that it could get very overwhelming. It takes a lot of dedication to learn abstract algebra, prove examples, get lost, and learning how to write a proof in the process. If the OP doesn't mind this though, he should go for it.
  8. Aug 11, 2011 #7
    Indeed, it might get overwhelming. But I feel like Pinter's book is the easiest introduction available, maybe it'll do the trick...
    The thing is that you'll have to learn proofs anyway, so now could be a good start. With only a proofbook, or while studying another course...
  9. Aug 12, 2011 #8
    Another book to look at is Survey of Modern Algebra by Birkhoff and MacLane:
    The authors write in the introduction that the book is written so that people with only high school maths can understand it. I think it is a very good book for such people, who only know school mathematics, and is a very enticing book. Books for university students can be too abstract and boring, this one isn't. It tries to relate the material to mathematics you learnt in school. Btw, don't get their book simply titled Algebra, because that's too advanced.
    Last edited by a moderator: Apr 26, 2017
  10. Aug 12, 2011 #9
    I think it's perfectly alright to try to jump in and look at mathematics above your level to get a few ideas about it, even if you don't understand everything. I've found this to be motivating. Dummit and Foote is pretty easy, I took a course using it with zero formal background in proof writing and had no difficulty.
  11. Aug 12, 2011 #10
  12. Aug 12, 2011 #11


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    If you go with Pinter, see the Amazon review by tech book guy that suggests you buy a non-Dover edition because of the small print; and have fun reading the reviews!
  13. Aug 12, 2011 #12
    Some suggestive comments:

    No. 4 has two, maybe three, chapters on algebra, the rest of the book being on other things. Also, it looks more like a reference book than a book that's going to teach you all those things, but I haven't read it so I can't be sure. It's also expensive.

    No.s 2 and 3 are similar, so you can get one or the other, you don't need both.

    No. 5 covers only linear algebra and not other parts of abstract algebra. It may go too far into this one topic if you just want a glimpse at what modern algebra is all about.

    No. 6 and 7 are the same book but different publishers. You have to ask yourself whether the $100 price difference is justified.

    Since you are reading for your own enjoyment, what is most important is whether the book appeals to you or not. Trying reading the first bits of the book on google books or wherever, in the library if they are there, and see which appeals to you the most. Though I see no.1, my suggestion, is not avaible on google books.

    If you still cannot decide, here is my suggestion. Get one (or both if you really want to) of no.s 2 and 3, and one (or both) of no.s 1 and 6.
    If you are on a budget, get no.s 2 and 6, which is the cheapest combination.

    Good luck
    Last edited: Aug 12, 2011
  14. Aug 12, 2011 #13
    thank you. and good idea, I will check them out on google books.
  15. Aug 12, 2011 #14


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    I have a copy of the Dover edition and I have to disagree with that reviewer. The font doesn't seem to me to be any smaller than in any other Dover book. It's perfectly readable.
  16. Aug 12, 2011 #15
    I guess I'll be reading "How to Prove it A Structured Approach" and "A Survey of Modern Algebra " because I found free online PDF's of both books! Woo hoo! lol
  17. Aug 13, 2011 #16
    I am also interesting in reading Maths and I want to learn Abstract Algebra,I am reading the linear algebra part in Tom M Apostol calculus Vol.2.What do you recommend me to do ?
  18. Aug 13, 2011 #17
    Honestly, if you're already used to proofs and linear algebra, I'd go ahead and see if you can get through some of Michael Artin's Text "Algebra", it's hands down the best abstract algebra book IMO. Exposition is clear and concise and well motivated.
  19. Aug 15, 2011 #18
    I agree, my Dover version of Pinter is also perfectly readable. One of my favorite Dover books.
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