Introduction to Abstract Algebra

[NukeEng101]

I was wondering if anyone could give me any links or an introduction to abstract algebra. I know that abstract algebra is a tough concept to understand (at least for some people, but it varies from person to person). If anyone could help with the basics of it would be greatly appreciated.

 

[NukeEng101]

Also I believe that linear algebra comes before abstract algebra, so if someone could help give a brief introduction to that as well would be great. I am currently a high school senior taking calculus 2 at a college level so I am very interested in the higher math classes and I know there are n number of courses I should probably take beforehand but I am just curious about the subject.

 

[micromass]

Abstract algebra is an incredibly fun thing, and I wouldn't consider it difficult. There are way more difficult things out there than groups.

What abstract algebra tries to do, is to generalize the arithmetic rules to a broader concept. For example, you (should) know how to solve 2x+1=4. Well, the techniques that you use for this are not limited to R. Instead, abstract algebra generalizes these techniques to groups and rings. So in a way, a group (or a ring) is the most general concept in which you can solve equations. Then a course in abstract algebra will try to find links between several groups and they will prove very beautiful theorems about them.
Now, why is this all useful? Because, in the end, we can prove things in this new framework which we couldn't prove normally (or whose proof would be very difficult). One example is the unsolvability of the quintic. You've undoubtely seen how to solve quadratic polynomials. Well, the same is possible for cubic and quartic polynomials, but not for quintics.

You need almost nothing for abstract algebra (but of course: the more you've seen, the better you'll be at it. Not because of the background information, but more because of "mathematical maturity"). You'll only want to be able to handle "abstraction". I'll give an analogy: when you were 6, you learned how to add two numbers, and that was it. But after a few years, you added letters in your equations. You "abstracted" the concept. This is what abstract algebra does... a lot. So you will have to accept abstraction.
You'll also do a lot of proofs in algebra. If you don't like proofs, then abstract algebra is not for you. (or, generally, mathematics is not for you)

I wouldn't say that you need linear algebra for abstract algebra. I actually argue that it's the other way around. But colleges give linear algebra first because it's applications are very important to all kinds of fields.

In short: linear algebra studies systems of equations. It considers the most general place where you can solve equations: the vector space. It also deals with matrices (and it are these that you need in abstract algebra).

Now, a very, very, very good book in abstract algebra is "a book of abstract algebra" by Pinter. (https://www.amazon.com/dp/0070501386/?tag=pfamazon01-20). They claim in the comments that even a ninth-grader can understand it. This is very true, the book is THAT easy. You'll only need to work a little for it (i.e. make all the exercises).
There will be some matrices in the book, but they're not fundamental. They can easily be skipped as they only arise in the exercises.

 

[NukeEng101]

Okay thanks, I'll definitely check out the book. Thank you as well. I like to do the proofs and anything of the sort. I just love how precise but yet beautiful how mathematics can be at times and I just wanted to broaden my horizons a little bit.

 

[Deveno]

linear algebra versus abstract algebra...hmm. it's like this: at first, when exposed to mathematics, you learn arithmetic. later, you learn that the laws of arithmetic can apply to numbers that you aren't even sure "which" numbers they are, but by manipulating the rules you can often discover a lot about these "unknown numbers". abstractly, linear algebra is just a small subset of abstract algebra (a "nice" special case, much like continuous functions are "nice" special cases of functions). but linear algebra also has an arithmetic, the arithmetic of matrices.

so often linear algebra is presented first, by way of its arithmetic, matrices, because you can get your hands on a matrix, and start doing actual calculations. and pairs of numbers (x,y) in the plane, or triples (x,y,z) in space are things which seem "real" and concrete, so the vector spaces R^2 and R^3 are used extensively as examples, they are familiar.

abstract algebra, on the other hand, is hard-core to the rules straight-away. it turns out that several things that look different are treated "just the same" because they obey the same rules. one doesn't usually think of the integers mod 4, and {1,-1,i,-i} as the same thing (they certainly aren't the same set), but an algebraist doesn't care, to him/her, they are "equal up to isomorphism", and that's all that matters.

abstract algebra is unifying, it allows us to recognize "sameness" in things we thought of as "different". for example, it makes it clear why "factoring polynomials" is similar to how we find the "greatest common denominator" when we add fractions.

there is a fairly simple introduction to group theory at this site: http://dogschool.tripod.com/

i rather like this site for linear algebra, it is not too abstract for a first-time exposure:
http://tutorial.math.lamar.edu/Classes/LinAlg/LinAlg.aspx

(it also has some worked-through problems and solutions, always a plus).

 

[DrWillVKN]

Will studying real analysis help with understanding the concepts in abstract algebra (excluding the fact that it improves your proof writing skills)? Or are the concepts completely different?

 

[micromass]

Will studying real analysis help with understanding the concepts in abstract algebra (excluding the fact that it improves your proof writing skills)? Or are the concepts completely different?

No, not at all. Of course, somebody who knows real analysis will be more able to study abstract algebra, but that's because of mathematical maturity, proof-writing skills,...). But if your goal is to study abstract algebra, then you don't need real analysis at all!

 

[Deveno]

real analysis relies heavily on the structure of the real number system, which is fairly "specific". the real numbers are a complete archimedian ordered field, which is a very special type of algebraic object. so you're not going to get much exposure to more general algebraic objects.

what real analysis IS good for, is as a pre-cursor to topology. after a solid real analysis course, you should be comfortable with the idea of a metric space, for example.

both topology and abstract algebra come together in algebraic topology, where the methods of algebra, and the methods of analysis can be used to "transfer" information of one kind to the other.