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Integer-valued polynomials July 21, 2009

Posted by Akhil Mathew in algebra, Problem-solving.
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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, I July 19, 2009

Posted by Akhil Mathew in algebra.
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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.