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I have listed 3 textbooks i have in my mind to buy

  1. Herstein - Topics in Algebra
  2. Artin - Algebra
  3. Lang - Undergraduate Algebra

Unlike Lang's Algebra is the most traditional abstract algebra text for graduates, I see there is no outstanding abstract algebra text for undergraduates.

I'm familiar with set theory,real and complex analysis,topology and some 'intuitive linear algebra'. So what would be the best among these three texts? or is there a traditional abstract algebra text? I'm NOT asking you to suggest me a book you think is the best, but to tell me the most traditionally using text book please.

Plus, there is another question. Generally, linear algebra is taught in freshman course and Abstract algebra is taught in junior course. WHY? In my opinion, one should study abstract algebra first, then linear algebra later to grab genuine concepts of those.

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closed as not a real question by YACP, Micah, Henry T. Horton, Brandon Carter, Ron Gordon Feb 5 '13 at 5:15

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It depends on what you call linear algebra. Matrix manipulation (calculating determinant, conjugation, row reduction, etc) does not constitute the abstract algebra portion of linear algebra. –  Calvin Lin Feb 5 '13 at 0:42
    
@CalvinLin I believe linear algebra is the study of objects written in 'abstract algebra language'. Am I wrong? –  Orgasmus Feb 5 '13 at 0:47
    
Your question is not well-defined. What does it mean to be "most traditional"? Are you asking for somebody to give you statistical data on which text is used the most out of all undergraduate institutions teaching abstract algebra, worldwide? I doubt anybody knows the answer to that. –  Christopher A. Wong Feb 5 '13 at 0:49
    
I would say that field theory " is the study of objects written in 'abstract algebra language'" but I wouldn't call it linear algebra –  Belgi Feb 5 '13 at 0:50
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I love Abstract algebra by Dummit and Foote, it at least needs to be on the list –  Belgi Feb 5 '13 at 0:51

1 Answer 1

up vote 4 down vote accepted

As per my comment above, I'm not in a position to answer your first question about what text is "most traditionally used". However, I can certainly take a stab at your second question.

Linear algebra is taught before abstract algebra for two main reasons.

  • Lots of people need to know linear algebra: scientists, engineers, programmers, mathematicians. Not as many need to know abstract algebra, at least not in as direct a manner as linear algebra. In the typical academic course hierarchy, general purpose classes are taught before specialized courses, so consequently linear algebra precedes abstract algebra.

  • Linear algebra is simply more concrete than abstract algebra. The mathematical ideal is that we start with the foundations, the most general theory, and then work our way up by building up additional structure, additional theory. But you can't really do that when you're first learning mathematics, especially the elementary ideas. When one first learns abstract algebra, almost every student will ask "What are some interesting groups?" and "What are some interesting rings?" A knowledge of linear algebra eliminates a large part of the need for a course in abstract algebra to deviate in order to motivate these structures, since students will more readily see abstract algebra as a natural generalization of topics they are already comfortable with.

Without truly answering your first question, I can at least note that Artin's algebra book takes a very different approach to the other two, so in some arbitrary sense it is the least traditional. Artin prefers to introduce its readers first to matrix groups, because they are more concrete, and then only later does he proceed to the more abstract notions.

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+1: That's not to mention that algebra is infinitely harder than linear algebra... –  gnometorule Feb 5 '13 at 1:02
    
I like your second reason. –  Orgasmus Feb 5 '13 at 1:10

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