# Topics in Finite Fields

## 642:587, Fall 2019

## Course Info

Instructor: Swastik Kopparty (swastik.kopparty@gmail.com)

Class Time and Place: Mondays and Thursdays 10:20 am - 11:40 pm, in Hill 425

Office Hours: Thursdays 3pm-4pm in Hill 432

Prerequisites: undergraduate level abstract algebra, combinatorics, probability, graduate level 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
- Algorithms
- Applications to Theoretical Computer Science, Combinatorics and Number Theory

There will be 3-4 problem sets.

## Homework

- HW 1 due Thursday, October 3

## Notes

## Lecture Schedule

- September 5: finite field basics: existence, uniqueness, construction, etc.
- September 9: finite field basics continued
- September 12: the Fourier transform
- September 16: NO CLASS (makeup to be scheduled)
- September 19: Gauss sums, simple applications of Gauss sums
- September 23: NO CLASS (makeup to be scheduled)
- September 26: the finite field Waring problem
- September 30: character sums with polynomial arguments: the Mordell bound
- October 3: Kloosterman sums, an application
- October 7: the Weil bounds, statements and an application
- October 10: proof of the Weil bound for quadratic residue character
- October 14: factoring univariate polynomials
- October 17: deterministic univariate polynomial factoring over small fields,
randomized primality testing
- October 21: deterministic primality testing
- October 24: the Cauchy-Davenport theorem
- October 28: baby sum-product theorems
- October 31: proof of the sum-product theorem
- November 4: proof of the sum-product theorem, contd.
- November 7: sumset inequalities
- November 11: the Balog-Szemeredi-Gowers theorem
- November 14: bounds on Gauss sums for small subgroups
- November 18: zeroes of polynomials, the Chevalley-Warning theorem
- November 21: more on zeroes of polynomials
- November 25: polynomials taking values in a subfield
- TUESDAY November 26:
- December 2: BCH codes
- December 5: linearized polynomials, subspace polynomials
- December 9: applications to integers