Recent content by altcmdesc

Great. A somewhat related topic is Algebraic K-Theory... so what about schools for that? Since it isn't a huge subject you don't really hear about schools having math departments which are strong in the field..

I'm finding myself interested in studying something like Algebraic Geometry or Algebraic Topology for graduate school, and I'm just wondering what schools are out there at have strong programs in these areas? I already know about Columbia, Michigan, Texas, and Utah for AG, and about UChicago for...

I've had a graduate course in algebra, so I'm familiar with the basics of modules and Galois theory -- both of which I find interesting. It's hard for me to be more specific and say exactly what I like about algebra/category theory, but I guess it's the level of abstraction and the way that one...

Nothing? Surely someone can give a little input from their perspective.

Just looking for a little advice on subfields of math to study up on. I'm asking since it's hard for me to know what some subfields are like looking at them from a "bottom-up" perspective. I'm a huge fan of algebra/category theory, but I feel like some of the problems can be a bit uninspired...

Proving an "or" statement What is the general procedure when proving a statement like "A implies B or C"? Is it more common to assume A and then split into cases (i.e. case 1: A implies B, case 2: A implies C, therefore A implies B or C)? Or is it more common to assume A and that one of B or C...

What's the difference (if any) between a direct sum and a direct product of rings? For example, in Ireland and Rosen's number theory text, they mention that, in the context of rings, the Chinese Remainder Theorem implies that \mathbb{Z}/(m_1 \cdots...

I'm looking for a good number theory book which doesn't hesitate to talk about the underlying algebra of some of the subject (e.g. using group theory to prove Fermat's Little Theorem and using ring theory to explain the ideas behind the Chinese Remainder Theorem). I'm still an undergraduate, so...

That's the conclusion I reached as well, but I figured that it was too simple of an answer to a challenge problem..

Sure: http://www.math.harvard.edu/~tomc/math25a/challenge.pdf It's from a class held Spring 2005. There are other challenge problems posted on the course webpage which are interesting as well.

This isn't a homework question or anything, but I came across this challenge problem posted on a Harvard Math 25a webpage and I'm wondering what the solution to it is since no solution is posted on the page. Suppose that f\colon \mathbb{Q} \to \mathbb{Q} satisfies f'(x) = f(x) for all x \in...

It acts on itself by conjugation, right? Does that action apply to sets with less than n! elements as well by taking an element a in the finite set and multiplying it by pap^{-1}?

My point is that if the vector space is over the reals, for example, not every characteristic polynomial has roots (e.g. x^2+1), so if T^m(V) isn't over an algebraically closed field, we don't necessarily have that the restricted map has an eigenvalue...right?

So, essentially, S_n can act on any set of more than n elements by just permuting n of the elements and leaving the rest alone?

Why must the restricted map have an eigenvector? I know where to go from there, but I can't seem to understand why that must be true since there isn't any assumption made that the base field of the vector space V is algebraically closed.