Finite Fields

Fall 2013

(642:587)

 

Course Info

Instructor: Swastik Kopparty (swastik.kopparty@rutgers.edu)

Class Time and Place: Mondays and Wednesdays, 5:00pm – 6:30pm, in Hill 425

Office Hours: Wednesday 3:30-4:30 (Hill 432)

Prerequisites: undergraduate level abstract algebra, mathematical maturity.

References: Lidl & Niederrieter (Finite Fields), Schmidt (Equations over Finite Fields), Tao & Vu (Additive Combinatorics).

 

 

Syllabus

 

This course will cover some important classical and modern themes in the study of finite fields. These will include:

·         Solutions of equations

·         Pseudorandomness

·         Exponential sums and Fourier techniques

·         Algebraic curves over finite fields, the Weil theorems

·         Additive combinatorics and the sum-product phenomenon

·         Many applications to combinatorics, theoretical computer science and number theory

 

There will be 2-3 problem sets.

 

Homework

·         Homework 1 (due September 25)

·         Homework 2 (due November 4)

 

Lecture Schedule

·         September 4: finite field basics (notes)

·         September 9: finite field basics, continued

·         September 11: finite field basics, introduction to Fourier analysis on finite abelian groups

·         September 16: more Fourier analysis, the Gauss sum (notes)

·         September 18: character sums over algebraic sets

·         September 23: the Waring problem

·         September 25: character sums with polynomial arguments (notes)

·         September 30: character sums with polynomial arguments, continued

·         October 2: polynomials over finite fields: basic properties

·         October 7: irreducible polynomials, zeta and L functions (notes)

·         October 9: irreducible polynomials in arithmetic progressions

·         October 14: (3 hour class) the Weil bound (notes)

·         October 16: the Weil bound, continued

·         October 21: (3 hour class) the Weil bound, continued

·         October 23: the Weil bound, continued

·         October 28: no class

·         October 30: applications of the Weil bound

·         November 4: (3 hour class) some nuggets of additive combinatorics

·         November 6: additive energy

·         November 11: sumsets

·         November 13: the sum product theorem

·         November 18: (3 hour class) additive character sums over multiplicative subgroups

·         November 20: counting integer solutions to polynomial equations, compactness

·         November 25: