Math 620 Notes - Fall 2009

Galois Closures of ring extensions:

1. An excerpt from a paper by Bhargava This is from ``Higher composition laws III: The parameterization of quartic rings," by M. Bhargava, Annals of Math, 159 (2004), p. 1329 -1360. This one of a sequence of four papers in the Annals by Bhargava which would be interesting to discuss at some point in the Galois seminar.
2. Notes from a Talk by Bhargava given in Leiden, in April 2009 These notes give some info on recent developments with Galois closures of rings. One open problems has to do with finding an appropriate generalization to rings of the field extensions one obtains by adjoining some number (but not all) of the roots of an irreducible monic equation over the ground field.

Polynomials which take on integral values on a specified set

1. An American Math Monthly article by Bhargava This is an introduction to the subject.
2. A Journal of Number theory article by Bhargava This article gives more details on the construction of generlized factorials
3. An article by De Marco and Rumely in Crelle's Journal This is a somewhat technical article about capacity theory and transfinite diameters, which is closely connected with the topic. I would suggest first having a look at section 2. For more relevant background, see the book by Rumely titled ``Capacity Theory on algebraic curves".

Last updated: 9/22//09
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