Spring Semester, 2001

Course Title:  Topics in Algebra, Equations over finite fields
Unique Number:  M390C (55320)
Time and place:  TTh 12:30-2:00 RLM 10.176
Instructor:  Felipe Voloch

Brief description:  We will study the classical topic of counting or estimating the number of solutions to (systems of) polynomial equations over finite fields. We will first review the basic theory of finite fields and study some elementary and combinatorial bounds, such as the Chevalley-Warning theorem and generalizations.

We will then move to the theory of curves over finite fields, prove Weil's analogue of the Riemann hypothesis and discuss some improvements to it. Time permitting, we will have an introduction to the statement of the Weil conjectures and the results of Dwork, Grothendieck and Deligne.

Class notes taken by students

The only requirement for this class, for registered students, is to take turns to write up notes in TeX that I will make available to all other students through this webpage. The notes are not meant to be in final form and I would encourage you to send me any corrections or comments.

These notes have now been put in a uniform shape (by Brian van der Ven) and edited to remove some glaring errors and misprints (by me). You can download the whole thing in one file pdf.

Prerequisite:  graduate algebra

Textbook: none