Filed under: Algebra, Math | Tags: abelian groups, algebra, commutative, group theory, Math
Wow, it’s been a long time since I’ve written anything on this blog. I’m taking algebraic topology and an algebraic number theory course this semester, and I started reading through Atiyah and MacDonald’s Commutative Algebra over the winter. So I thought I’d continue with a little algebra. The algebra we’ve done thus far has been highly noncommutative, for the most part — we investigated groups like free groups, symmetric groups, matrix groups, and dihedral groups in which the order of operations mattered. As you might expect, with abelian groups, the theory becomes much simpler, and the subject called “commutative algebra” is just the study of abelian groups with extra structure — something like a scalar multiplication, as in the case of vector spaces, or some other operation. But first, we need to understand abelian groups.
When talking about abelian groups specifically, we usually write them additively: the group operation applied to and is , and then we can build expressions like . The proof I give below is due to J. S. Milne, who in turn says it’s similar to Kronecker’s original proof. Of course, I’ve added more detail in places where I thought it was necessary, and taken it out where I thought it wasn’t. There are other, more common proofs, typically using matrices, but I find them unwieldy and inelegant.
Filed under: Uncategorized
It’s looking to be a pretty good year. I’m cooling my heels in the Philippines, where it’s nice and sunny but I don’t know practically anyone. All the more excuse to sit outside in the sun and do some math. My parents, who are awesome, gave me Stein and Shakarchi’s Princeton Lectures in Analysis, Volume I: Fourier Analysis. Volume III (Real Analysis) was the textbook for my analysis class this past semester, and I’d briefly skimmed Volume II (Complex Analysis) for a freshman year class, but I wanted to really sort of start from the beginning and get my hands dirty with some of the basic material of analysis.
The subject itself is pretty cool: Fourier series are a way of decomposing any periodic function (i. e. for all ) as an infinite sum of sines and cosines, and this sum generalizes to an integral called the “Fourier transform.” A good comparison is the Taylor series, which decomposes a function as an infinite sum of polynomials. Oddly enough, besides last semester’s analysis class, the only place I’d seen this was the same freshman class where we used Stein and Shakarchi II: you can give a nice algebraic summary by saying that the set of functions whose squares are integrable (“ space”) is a vector space with inner product , and the set of functions , and forms an (infinite) orthonormal basis for this vector space. Regrettably, you have to use the Lebesgue integral to do this, and since S&S want their series to be self-contained, they only use the Riemann integral. Which makes questions like “when does the Fourier series converge properly?” a whoooooole lot easier, but you kind of feel you’re missing the big picture. On the other hand, the books are very well written, lucid without being verbose, and the problems are usually pretty interesting. S&S I is geared on the easy side of things and is probably a good read for even interested high schoolers (and just like high school, there are tons of integrals to be done! woohoo…).
Probably the awesomest Christmas present I got was the sadly-out-of-print Mathematics Made Difficult by Carl E. Linderholm, written in the 70′s. As the title would suggest, the book is fed up with the never-ending attempts to make math seem easy or fun, and instead aims to present basic arithmetic in as difficult a way as possible. It’s an odd book, and you see why it’s out of print — it’s geared towards mathematicians and math students, defining the set of natural numbers, for example, as an initial element in the category of pointed monoids, but at the same time it’s possessed by a Groucho Marx-style wit and wordplay, and ends up making all of these wry observations about the philosophical issues of thinking about math that are usually consigned to pop-sci books.
“[One might object, to the proposition that , that] 2 units multiplied by 2 units is not 4 units but 4 squared units. But 4 squared is 16. Hence, .
“[To which I reply that] it is not at all obvious a priori that 16 is not exactly 4.”
And in the introduction, he starts a paragraph with “Consider a category…” and proceeds to give a long, rambling list of properties that category should be assumed to hold. Then he ends with “Then it is possible to think, for we have just defined a Boolean algebra.” Hilarious. (A Boolean algebra is a mathematical structure that looks like the class of sets with the operations of union, intersection, and complementation. Deductive logic itself basically has an underlying Boolean algebra structure, so the existence of one allows us to conclude that logic, and so thought, exist too.)
I’m also reading Infinite Jest for a book club-type thing with my friends, and helping my girlfriend put together a website. As someone who’s always been sort of interested in techy stuff like programming and web design, but never had any real reason to go out and do it, I cannot emphasize more the coolness of actually having somewhere to put one’s skills to the test and force oneself to learn.
But I guess the reason why I’m starting this post is that I need to pick another math book to read. I’ve fortunately got a pretty nice electronic collection, and I’d like to find a subject that’s been somewhat out of my path so far — algebraic geometry, number theory, and category theory are all looking good. Lethe Ar has a review of J. P. “Duh Bear” Serre’s Arithmetic here — I share his disdain and confusion for the subject but it looks like it might be an interesting read… Or before tackling those, should I brush up on my commutative algebra? I’m fine with groups and fields, pretty good with homological stuff, but always a little cautious with rings and modules. I’ve got Dummit/Foote and Atiyah/Macdonald, both of which I believe are pretty standard texts (Holomorphusion‘s been going through Atiyah/Macdonald, FWIW).
If any math person reading this has any recommendations, I’d love to hear them. And, as usual, if anybody’s curious about some topic, I always welcome suggestions. Happy New Year!