I know it’s cool. I know we wouldn’t have a zillion algebraists at my school if it weren’t cool. But dangit, while I was taking that class I had no motivation to do any of it. For the first time in my life, I couldn’t bring myself to care about a class. It just seemed so, for lack of a better word, abstract. Since then I’ve had a friend explain a bit about the relation between Lie groups and particle physics, and I know there’s more neat stuff like that, I just don’t know what it is. Are there any good books/journals I could read? Help me care about modern algebra!
Since the responses will be a matter of opinion, I’ll send this thread to IMHO.
Are you the kind of person who likes something only if it’s useful? If so, probably nobody can help you.
No Achernar, I definitely wouldn’t say that’s the case. And I hope this doesn’t sound like a “Dude, math blows” sort of rant. I love math, both theoretical and applied. I just have a hard time seeing what draws so many people to the subject of abstract algebra.
Okay, that’s fine. I’m honestly trying to help you here, and I believe that your intentions are good. Can you think of something which is extremely theoretical which you enjoy? That may be a good starting point.
Most cryptographic theory was developed from abstract algebra. A decent number of algebraists go on to careers in the NSA for that reason.
If you like math in general, but don’t like algebra, then it might just be that you don’t like algebra. Each branch of math has a different flavor, and people have different tastes. I definitely don’t like analysis that much, and though I prefer algebra, my favorite is logic and computation theory. Do you have a favorite kind of math?
I’m just a junior at university, so I’ve barely started getting beyond the basics. The small bits of topology I’ve had (especially knot topology) I’ve found really interesting. I really like my real analysis class I’m in right now, although I don’t know how theoretical you’d consider that to be. I enjoy logic a lot, and number theory too.
I think my favorite subject so far has been complex systems. Unfortunately the only class I’ve had on that only had a bit of theory, so I’ve only been able to absorb bits of measure theory stuff in my spare time. My least favorite class was in pdes. We spent an entire semester learning about a trillion ways to solve different versions of the wave equation. The engineering students dug it, but the rest of us were bored stiff.
I guess ultrafilter, it could be that I just don’t care for algebra. It seems like a lot of the math I enjoy is similar to the kinds of logic puzzles and geometrical proofs I used to love to do when I was a kid, and algebra (in my limited experience) just hasn’t been like that. I also didn’t have the best professor in the world, and that gave me a bit of a distaste for the subject too. Or maybe I just hang around too many physics majors. 
Anyway, thanks everyone bearing with me.
On a slightly related note, which algebra textbooks would you recommend? I used Herstein, but I’d really like another source to study for the gre and eventually my quals.
I used Herstein, but I had a good professor, so I didn’t need to read the book much. A few of the reviewiers of Herstein on Amazon mention other books that they would prefer, or you can check out Dover’s web site, where they offer good math books for less than $20.
IME, algebra is the closest to the geometry-type stuff from way back when, but that may be because of the professor I had. Different profs, different impressions, y’know?
Ooh, thanks for the Dover link. I thought they just made sticker books! I guess I’ll do some reading this summer and maybe I’ll change my tune. Thanks for your help.
Sorry it took so long, but I’ve spent some time thinking about this, and I think that there are three reasons why I like Abstract Algebra, and none of them is that it’s useful.
First, it can lead to some very fundamental understandings of some very important sets, like Integers and Reals. You get to know their defining and incidental properties rather well.
Second, look at this picture I made. It’s all the elements of the ring Z[sub]13[/sub]*, with units in blue all non-units in red. It’s beautiful. I love being able to make beautiful things. For me, Algebra shares this in common with Fractal Geometry. (Incidentally, if you’re wondering about Number Theory, the picture is also lattice points (x, y) such that 13 divides x[sup]2[/sup] + y[sup]2[/sup].)
Third, and this is probably most important for me, many of the proofs and problems in Algebra require a certain elegance. For me, Algebra shares this in common with Plane Geometry, as opposed to, say, Calculus, in which many of the problems were very straightforward. As an example, here’s a problem from the Algebra book I used (Contemporary Abstract Algebra (4th ed.) by Joseph A. Gallain - I really like this book). I think this problem is quite easy, but shows what I’m talking about:
3.27. Suppose a group contains elements a and b such that |a| = 4, |b| = 2, and a[sup]3[/sup]b = ba. Find |ab|.
Does |a| represent the order of the element a? If so…|ab| = 2, correct?