**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.
- "p-adic Numbers, p-adic Analysis, and Zeta-Functions" by Neal Koblitz. 2nd edition. Published by Springer-Verlag, 1984. 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.

Rami's home page.

Last modified:
October 24^{th}, 2001 |