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## Integer-valued polynomialsJuly 21, 2009

Posted by Akhil Mathew in algebra, Problem-solving.
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6 comments

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 ${P(X) \in \mathbb{Q}[X]}$ which takes integer values at all sufficiently large ${n \in \mathbb{N}}$. What can we say about ${P}$?

Denote the set of such ${P}$ by ${\mathfrak{I}}$. Then clearly ${\mathbb{Z}[X] \subset \mathfrak{I}}$. But the converse is false.

## Why simple modules are often finite-dimensional, IJuly 19, 2009

Posted by Akhil Mathew in algebra.
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3 comments

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 ${A}$ be an associative algebra with identity over an algebraically closed field ${k}$; suppose the center ${Z \subset A}$ is a finitely generated ring over ${k}$, and ${A}$ is a finitely generated ${Z}$-module. Then: all simple ${A}$-modules are finite-dimensional ${k}$-vector spaces.

We’ll get to this after discussing a few other facts about rings, interesting in their own right.