What exactly is Abstract Algebra?

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Discussion Overview

The discussion revolves around the nature of abstract algebra, its relationship to modern algebra courses, and the prerequisites typically required for studying the subject. Participants share their experiences and expectations regarding the course content and difficulty level.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants question whether "Modern Algebra" is synonymous with abstract algebra, citing differing opinions on course content.
  • One participant mentions that their teacher referred to abstract algebra as "Rings n Things," suggesting a focus on ring theory within the subject.
  • Another participant lists a variety of courses they believe are prerequisites for abstract algebra, including calculus and linear algebra, while expressing uncertainty about their universality.
  • Some participants argue that abstract algebra can be approached without extensive prior knowledge, emphasizing that it often starts from fundamental concepts.
  • One participant asserts that understanding and constructing proofs is crucial for success in abstract algebra, highlighting the need for abstract thinking.
  • Another participant shares their experience of taking a first course in abstract algebra, noting that it covered basic concepts like groups and binary operations but did not delve into rings and fields.
  • There is a discussion about the necessity of linear algebra for understanding abstract algebra, with differing views on its relevance to the foundational concepts of groups and rings.
  • One participant suggests that a basic understanding of functions and induction may suffice for studying abstract algebra, indicating variability in course expectations.
  • Concerns are raised about the varying levels of rigor in courses labeled as linear algebra, with some being more advanced than others.

Areas of Agreement / Disagreement

Participants express a range of opinions regarding the prerequisites and foundational knowledge required for abstract algebra, indicating that there is no consensus on what is necessary to succeed in the subject.

Contextual Notes

Participants highlight the variability in course content and prerequisites across different institutions, suggesting that definitions and expectations may differ significantly.

RJC
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I'm 17 and my high school has no other math courses to offer me. At a local college there is a course called "Modern Algebra" and I was wondering if it was the same thing as abstract algebra. I asked my math teacher about it, and he said it was the hardest math class he took; he used to call abstract algebra "Rings n Things." And also, what are the prerequisites to this course?
 
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RJC said:
I'm 17 and my high school has no other math courses to offer me. At a local college there is a course called "Modern Algebra" and I was wondering if it was the same thing as abstract algebra. I asked my math teacher about it, and he said it was the hardest math class he took; he used to call abstract algebra "Rings n Things." And also, what are the prerequisites to this course?
I haven't taken it yet, but I am also 17 and in the same position. I don't know if the prereqs are the same everywhere, but here you usually take single variable calc, multivariable calc, linear algebra, differential equations, discrete math, advanced calc, and complex variables. I'm sure you'd be fine without those last two though.
 
i haven't had about half of those, so i guess i won't be taking it!
 
An ounce of mathematical sophistication is probably all that's required for your average introductory abstract algebra course. I seriously doubt all those courses listed by apmcavoy (except linear algebra) are actual prereqs for a course in abstract algebra at any university, seeing as (at this level, anyway) they are totally irrelevant for the subject at hand (not that they can't be used to acquire the previously mentioned mathematical sophistication, though).
 
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Just to reiterate Muzza's point...

Abstract Algebra is a very basic subject: in it you would pretty much start from scratch and work from there. It generally doesn't build upon previous knowledge. (but the examples might!) In fact, it's one of the subjects that is used to a varying extent in just about every other subject.

The catch is that you have to be able to understand and construct proofs fairly well. Being able to think abstractly doesn't hurt. (But hopefully that can be taught!) That's why it's usually encountered late in the undergraduate cirriculum. (Or so I think)
 
I took a first course in Abstract Algebra last year, assuming it would be similar, it doesn't get very far. It covers the basics, defining binary operations, looking at groups, subgroups and some of their basic properties, that sort of thing. We never got as far as Rings and Fields. Without a prerequisite of Linear Algebra there is no way you'd get too deep into it.

If you are willing to keep an open mind to the material you will be taught and have a reasonable level of confidence with your mathematics, than you should be able to easily deal with it.
 
Linear Algebra is just basic college algebra, right?
 
RJC said:
Linear Algebra is just basic college algebra, right?

If abstaract algebra is rings 'n' things, then linear algebra is vector spaces 'n' junk. Actually you relaly don't need any linear algebra to understand the basic concpets of groups, semigroups, rings, etc all you really need is basic algebra.

After all a vector spaces (at least when first introduced) rely on the concepts of fields and abelian groups.
 
  • #10
look at a book like herstein's topics in algebra. or whatever it is called now. you do not need anything to study abstract algebra except a biasic understanding of injective and surjective functions between sets. maybe not even that if the course covers that too, as many do, and maybe induction.

linear algebra is logically a chapter within abstract algebra.

many different courses are taught with the same names, at various levels, many quite watered down. linear algebra can refer to a trivial course in which people only learn to manipulate matrices, or it can be a course on rings and modules, exact sequences, and normal forms for linear transformations and examples such as lie groups, lie algebras, and group representations.
 

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