Number Theory at CMU

Welcome to the course page for Number Theory at Carnegie Mellon University! This will be the main page for the course, and all updates will be posted here.

TENTATIVE OFFICE HOURS: MONDAYS 9-11AM, THURSDAYS 11-2PM

TEST 1 IS SET FOR WEDNESDAY, FEBRUARY 28, 2018--here're the spec on the exam

TEST 2 IS SET FOR WEDNESDAY, APRIL 11, 2018

Syllabus: click here

Textbook: click here. We will be using online notes of Pete L. Clark, which were developed when he taught a similar course at the University of Georgia.

Our course will consist of "modules", posted below. The modules will be a combination of lecture-notes/homework/extra-optional-reading.

Homework:

• • Due (by) Friday, January 26:

    • From Introduction/Infinitude of Primes: Exercise 1, 2 and 3.

    • From Divisibility: All exercises (13 and 14 are the tough ones).

  • Due Friday, February 2:

    • From Divisibility Tests: All exercises

    • From Wilson's Theorem: All exercises

  • Due Friday, February 9:

    • From Quadratic Reciprocity (Round One and Two): all exercises

  • Due Friday, February 16:

    • From Cyclotomic Polynomials: all exercises

    • From Euler phi function: all exercises

  • Due Friday, February 23:

    • From Multiplicative functions: all exercises

    • From Irreducibility Meets Multiplicative: all exercises

  • Due Monday, March 5 (I know HW is usually due on Friday, but you've got a test--you're welcome):

    • From Intro to Continued Fractions: all exercises

    • From Applications of Continued Fractions: all exercises

  • Due Friday, March 23

    • From Pick's Theorem: all exercises

    • From Minkowski's Theorem: all exercises

  • Due Friday, March 30

    • From Quadratic Forms: all exercises

  • Due Monday, April 16 (I know it's usually due on Fridays--test, though)

    • From Pythagorean Triples: all exercises

    • From Intro/Group Law: all exercises

  • Due by Monday, April 23 (I know...Carnival...no class...y'all better live it up)

    • Torsion Points--all problems

    • ECs over Finite Fields: all problems

Modules: see below. Note that the homework exercises are embedded in the modules. Order may change.

• Introduction/Infinitude of Primes

• Divisibility (note exercise 12 needs to read F_{gcd(m,n)}).

• Divisibility Tests

• Wilson's Theorem

  • Quadratic Reciprocity

  • Round One of Quadratic Reciprocity

  • Round Two of Quadratic Reciprocity

• Cyclotomic Polynomials and Irreducibility

  • Multiplicative Functions:

  • Euler phi function

  • Multiplicative functions in greater generality

• Irreducibility Meets Multiplicative: Counting Irreducible Polynomials Over a Finite Field

  • Continued Fractions

  • Introduction to Continued Fractions

  • "Applications"

  • Geometry and Number Theory

  • Pick's Theorem

  • One day/fun day (videotaped lecture pre Spring Break!)

  • Minkowski's Theorem

  • Quadratic Forms: the sum of squares and solutions for studying

• Pythagorean Triples and Fermat's Last Theorem

  • Elliptic Curves

  • Introduction/Group Law

  • Application: Congruent Number Problem

  • Torsion Points

  • ECs over Finite Fields; Birch and Swinnerton-Dyer

  • L-functions

  • Zeta(2) via Calc 2 and Diff Eq

  • Motivation and "Character Development" (still under development...)

  • Proofs of the Infinitude of Primes

  • Fermat's Christmas Theorem meets L-functions

  • Chevalley-Warning (one day, last day, fun day?)