Integer-valued polynomials July 21, 2009Posted by Akhil Mathew in algebra, Problem-solving.
Tags: integers, polynomials
So, in a break from the earlier series I was doing on Lie algebras, I want to discuss a very elementary question about polynomials. The answer is well-known but is interesting. It would make a good competition type problem (indeed, it’s an exercise in Serge Lang’s Algebra). Moreover, ironically, it’s useful in algebra: At some point one of us will probably discuss Hilbert polynomials, which take integer values, so this result tells us something about them.
We have a polynomial which takes integer values at all sufficiently large . What can we say about ?
Denote the set of such by . Then clearly . But the converse is false.
Why simple modules are often finite-dimensional, I July 19, 2009Posted by Akhil Mathew in algebra.
Tags: algebra, Hilbert basis theorem, Nullstellensatz, polynomials, simple modules
Today I want to talk (partially) about a general fact, that first came up as a side remark in the context of my project, and which Dustin Clausen, David Speyer, and I worked out a few days ago. It was a useful bit of algebra for me to think about.
Theorem 1 Let be an associative algebra with identity over an algebraically closed field ; suppose the center is a finitely generated ring over , and is a finitely generated -module. Then: all simple -modules are finite-dimensional -vector spaces.
We’ll get to this after discussing a few other facts about rings, interesting in their own right.