The crux of abstract algebra, I need help

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edahl
Posts: 61
Joined: Wed May 13, 2009 12:15 pm UTC

The crux of abstract algebra, I need help

Postby edahl » Thu Mar 18, 2010 3:22 pm UTC

Hello
I'm currently taking a course on abstract algebra using Fraleigh's wonderful book on the subject. Now, for reasons other than laziness I haven't been able to attend classes very much. I still want to finish class and I intend on working hard to get there, but I need to catch up on the big picture I'd normally get from attending class. That's the only way I'll know what should stick and what to think about so that working through problems and forming an intuition can occur in more of a context.

It's probably a lot to ask, but what I'd love to hear from you guys is pointers as to what is what in abstract algebra. What's the big picture, the important theorems, definitions, good ways to think about things, what sections and concepts in the book to pay extra attention too, perhaps shorter and better books on the subject, lecture notes, perhaps good assignments to get me thinking about this stuff the proper way.

To get an idea of the span of the subject and where I'm at, the course covers sections 0-6, 8-11, 13-23, 26-27, 29-31, 36-37, 45, 48-51, 53, 55-56, (I tried to find a PDF-preview of the Contents pages, but I sorrily couldn't find it, but I'm guessing this is a book a lot of people have) and we have been going 0->17,36,37,18->20 in that order. I had no problem with this course before I fell off for a few weeks for personal reasons. I feel I have not have had enough work on chapter 3 "Homomorphisms and Factor Groups" (13-17), and the rest follows :-)

I appreciate any insight!

thanks a bunch,
ed

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fractalchaos
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Joined: Wed Mar 17, 2010 5:36 pm UTC
Location: Ohio

Re: The crux of abstract algebra, I need help

Postby fractalchaos » Thu Mar 18, 2010 4:56 pm UTC

That book is so small, it's all pretty important. I used that book for two semesters of algebra about 5 years ago and we basically went through the entire book. It all builds on itself until you get to the final goal at the end: proving that there does not exist a formula to solve every quintic equation. I.E. there is the quadratic formula, there is a cubic formula, and a quartic formula, but it's not possible to come up with one that works for every possible quintic equation. The proof itself is not that complicated if I recall, but you have to know a lot of the stuff you learn along the way to get there. I would suggest just reading the book (skipping the starred sections) and doing some of the problems. Are you coving the whole book in your class?
Last edited by fractalchaos on Mon Mar 22, 2010 1:56 am UTC, edited 2 times in total.

Suffusion of Yellow
Posts: 42
Joined: Thu Jan 01, 2009 6:15 pm UTC

Re: The crux of abstract algebra, I need help

Postby Suffusion of Yellow » Thu Mar 18, 2010 11:57 pm UTC

Depends what in abstract algebra you're doing, honestly. If its focused on mainly group theory definitely know the Fundamental Homomorphism Theorem, Lagrange Theorem, and the Sylow Theorems.

edahl
Posts: 61
Joined: Wed May 13, 2009 12:15 pm UTC

Re: The crux of abstract algebra, I need help

Postby edahl » Sat Mar 20, 2010 10:33 pm UTC

Thanks! These are exactly the kinds of answers I'm looking for.

@Suffusion: I'm doing "the whole book". That is, group theory, ring theory, ..., galois theory. It's a first course thing.

It seems to me that in the end, all the stuff I'm learning this semester is stuff I absolutely HAVE to know in the end, and I probably will. Seems to weave together the carpet of A LOT many things. Pretty cool in my book.

Suffusion of Yellow
Posts: 42
Joined: Thu Jan 01, 2009 6:15 pm UTC

Re: The crux of abstract algebra, I need help

Postby Suffusion of Yellow » Sun Mar 21, 2010 12:31 am UTC

edahl wrote:Thanks! These are exactly the kinds of answers I'm looking for.

@Suffusion: I'm doing "the whole book". That is, group theory, ring theory, ..., galois theory. It's a first course thing.

It seems to me that in the end, all the stuff I'm learning this semester is stuff I absolutely HAVE to know in the end, and I probably will. Seems to weave together the carpet of A LOT many things. Pretty cool in my book.


Abstract Algebra is pretty cool in my book too. Lagrange Theorem never gets the recognition it deserves; its an extraordinary powerful and completely unexpected theorem with a simple, four or five line proof.


As for Ring Theory, definitely know the Fundamental Theorem again, and make sure you really understand what a quotient ring is, that's usually a trouble spot


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