WEEK LECTURES NOTES HOMEWORK DUE Week 1 01/21, 01/24 1/21 Notes Fermat's Last Theorem (statement, history), Parametrization of Pythagorean Triples, Plimpton 322 1/24 Notes Parametrization of Pythagorean Triples Cont'd Due 1/31 Week 2 01/28, 01/31 1/28 Notes Pythagorean Variety, Rational Points represented up to rational scaling by integer ones, Real points up to scaling are a circle. Pythagorean triples parametrize rational points on the circle. Start of proof of Fermat (to exponent 4). 1/31 Notes Fermat's Last Theorem (n=4), Proof, Fermat Descent, Quadratic Forms, Sums of Two Squares Due 2/07 Week 3 02/04, 02/07 02/04 Notes Sums of two squares, modular arithmetic, Fermat's Little Theorem, proof 02/07 Notes Euclidean Algorithm, Linear Equations in Integers Due 2/14 Week 4 02/11, 02/14 02/11 Notes Proof of Euclidean Algorithm, Groups/Rings/Fields, Ideals, Z/n is a field iff n is prime, Multiplicative Inverses, Units, Efficient Exponentiation, Euler totient function, Z is a Principal Ideal Domain 02/14 Notes Z is a Principal Ideal Domain, Euclidean Algorithm in Z[i] ("Gaussian Integers") Due 2/21 Week 5 02/18, 02/21 02/18 Notes Division Algorithm in Gaussian Integers Z[i], ord_p(n), Fundamental Theorem of Arithmetic 02/21 Notes Norm forms, lattice points on circles, Proof of Fundamental Theorem of Arithmetic, Failure of Unique Factorization in Z[sqrt[5]i], Order of a prime in a number, Group of units, Primes in general rings Lecture Exercises + Ch. 0 Exercises 3, 4, 12 Due 2/28 Week 6 02/25, 02/28 REVIEW 02/25 Notes Failure of Division Algorithm in Z[sqrt(-5)], Group of Units of a Ring, Associate Elements, Irreducible Elements, Prime Elements, Quadratic Residues/Nonresidues, How Many Residues, a^((p-1)/2)=+1 or -1, Efficient Algorithm for Finding Solutions to p=x^2+y^2 Lecture exercises only Due 3/06 Week 7 03/03, 03/06 MIDTERM 1 03/06 Notes Eisenstein integers, Z[w], w^2+w+1=0, as a lattice/ring, its norm form Lecture exercises Write up solutions to mistakes from midterm Due 3/13 Week 8 03/10, 03/13 CANCELLED 03/10 Notes Euclidean Domain, Polynomial Ring with Norm being Degree, Polynomial Division Algorithm, Euclidean Domain Implies PID, PID implies Noetherian Lecture exercises Due 3/27 Week 9 03/17, 03/20 SPRING BREAK Week 10 03/24, 03/27 03/24 Notes Review of Rings (Integers, Gaussian/Eisenstein, Polynomial), Ideals, Euclidean Domain (Division Algorithm, Norm), PID, Noetherian, UFD, units, associate, irreducible, prime elements. 03/27 Notes Rings, Ideals, Euclidean, Principal Ideal Domain, Noetherian, ascending chain condition, primes, irreducibles, ord_p, Unique Factorization Domain Due 4/03 Week 11 03/31, 04/03 03/31 Notes Eisenstein Integers, Numbers Represented by its Norm Form, x^2-xy+y^2, Discriminant, Definite/Indefinite, Homogenization, Finding Modular Cube Root of Unity 04/03 Notes Efficient algorithm for representing a prime = 1 mod 3 as the norm of an Eisenstein integer Due 4/10 Week 12 04/07, 04/10 REVIEW MIDTERM 2 04/07 Notes Review: Rings, Integral Domains, associate elements, units, irreducible, prime, Euclidean implies PID implies Noetherian implies UFD, polynomial division algorithm, Eisenstein integers, expressing prime as Eisenstein norm form Due 4/17 Week 13 04/14, 04/17 04/14 Notes Quadratic Forms (Binary), Degenerate, Split/Reducible, Definite/Indefinite, Discriminant 04/17 Notes Primitive Quadratic Form, a number being primitively represented, Integer General Linear Group, Spin Representation Due 4/24 Week 14 04/21, 04/24 04/21 Notes Binary Quadratic Forms, Equivalence, Same Numbers Represented 04/24 Notes Binary Quadratic Forms, Equivalence Classes, Determinant is Invariant, Roots, half-Hessian (Gram Matrix), Fractional Linear Transformations Due 5/01 Week 15 04/28, 05/01 04/28 Notes Reduction Theory of Binary Quadratic Forms (Definite), Upper Half Plane, Fractional Linear Action, Action on Roots is Inverse Action on Forms 05/01 Notes Reduced (Positive Definite) Quadratic Forms, Computing the Class Number, Gauss's Theorem: Finiteness of the Class Number Week 16 Office Hours: Tues 1-2 Week 17 05/13: Office Hours: Tues 1-2 FINAL EXAM Due: 5/13, 11 am