21-611 Topics in Algebra: p-adic analysis - Fall 2001 MWF 10:30

Instructor: Rami Grossberg
Office: WEH 7204
Phone: x8482 (268-8482 from external lines), messages at x2545
Email: Rami@cmu.edu
URL: www.math.cmu.edu/~rami
Office Hours: By appointment or whenever else you can find me.

Purpose. The p-adic integers where introduced by Hensel in the twenties as a tool to solve number-theoretic problems. Within a short time the deep theory of valued fields came to be. This theory consists of a combination of algebraic and some (easy) tools of analysis that is powerfull enough to resolve several deep number-theoretic problems.

Course description. The material to be covered: I will start with an elementary introduction to algebraic geometry and commutative algebra. Among the topics to be covered are: Zariski's topology, affine varieties, Hilbert's basis theorem and the Nullstllensatz. Basic facts about complete normed fields (most of the time we'll look at non Archimedean fields). The definitions of the p-adic rationals and integers. Ostrowski's characterizarion of non archimedean metrics. Hensel's lemma, "Newton's method of approximations". Local fields, and the completion of algebraic closure. Some basic properties of the p-adic versions of exp(x) and log(x) will be presented and some of the general theory of p-adic power series will be studied including the Weierstrass' preparation lemma. Eventually I will discuss B. Dwork's solution to the first of the Weil conjectures. Namely the finite analog to the zeta function is rational (quotient of two polynomials), in particular the number of roots of a polynomial (in all finite fields) can be predicted based on finite information.

Text: I will not use a text book. Much of the material can be found in the books listed below. I will assume the students have access only to the first two books.

• "Abstract Algebra" by D. S. Dummit & R. M. Foote. 2nd edition. Published by John Wiley & Sons, 1999. price comparison.
• "Local fields" by J. W. S. Cassels. Published by Cambridge Univ Press, 1986. price comparison.
• "A course in arithmetic" by J. P. Serre. Published by Springer-Verlag, 1990. price comparison.

Test Date: Will be announced.

Evaluation: There will be a midterm (one hour in class test), homework assignments, and a three hour final. These will be weighted as follows:

• The midtem will make up 20% of the grade
• The final will make up 40% of the grade
• Homework assignments will be collected every Monday, they make up 20% of the grade.

Prerequisites. Field theory (21-374) or Algebra I (21-610) and advanced calculus.

THERE WILL BE A TEST ON 10/24.

I will be leaving to Bogota on Tuesday morning (12/04/01) and will return on Sunday (12/16/01) afternoon. In case you have any mathematical or other problems please dicuss them with me before my trip or talk to Alexei Kolesnikov while I am out of town. I suspect that I will not have reliable access to email while away. I apologize for leaving at the end of the semester, but the conference is too important for me not to attend.