Topics in Finite Fields

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

• 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

• Finite field basics
• Fourier analysis
• Character sums with polynomial arguments

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