Date |
Section and Topic |
Read before class
|
Homework |
August 26 |
Organization and introduction
|
|
|
August 28 |
Unique Factorization in the integers
| Chapter 1; Sections 1, 2
|
HW 1 Due on September 9
|
August 30 |
Unique factorization in k[x]
| Chapter 1; Sections 2, 3
|
Chpt 1: 7, 9, 18, 21, 23, 25, 26, 35-38.
|
September 4 |
Unique Factorization in PIDs, Euclidean Domains
| Chapter 1; Sections 3, 4
|
|
September 6 |
Gaussian and Eisenstein Integers
| Chpt1; Sect 4. Chpt 2, Sect 1
|
|
September 9 |
Infinitude of Primes
| Chpt 2; Sections 1, 3
|
HW 2 due September 18
|
September 11 |
Arithmetic Functions
| Chpt 2, Section 2
|
Chpt 2; 1, 2, 6, 7, 8, 9, 10, 12, 15, 21, 26, 27
|
September 13 |
Growth of π(x)
| Chpt 2; Section 4
|
|
September 16 |
Congruences
| Chpt 3; Sections 1, 2
|
|
September 18 |
Chinese Remainder Theorem
| Chpt 3; Section 3, 4
|
HW 3 due Sept 30: Chpt 3; 1, 3, 5, 16, 18-21; Chpt 4: 1, 6, 8, 11, 12
|
September 20 |
Primitive Roots
| Chpt 4; Section 1
|
|
September 23 |
nth-power residues
| Chpt 4; Section 2
|
|
September 25 |
Quadratic residues
| Chpt 5; Section 1
|
|
September 27 |
Quadratic reciprocity
| Chpt 5; Section 2
|
|
September 30 |
Proof of quadratic reciprocity |
Chpt 5; Section 3
|
|
October 2 |
Algebraic numbers and algebraic integers
| Chpt 6, Section 1
|
HW 4 due Oct 14; Chpt 5: 1, 3, 16; Chpt 6: 2, 8, 11, 13.
|
October 4 |
The quadratic character of 2
| Chpt 6, Section 2
|
|
October 7 |
The quadratic Gauss sum
| Chpt 6, Section 3
|
|
October 9 |
Sign of the quadratic Gauss sum, a la Schur
| Handout
|
|
October 11 |
Sign of the quadratic Gauss sum, a la Kronecker
| Chpt 6, Sect 4
|
|
October 14 |
Introduction to Finite Fields
| Chpt 7, Sect 1
|
|
October 16 |
More on finite fields
| Chpt 7, Sect 2
|
Take home midterm due Oct 28
|
October 18 |
Quadratic reciprocity via finite fields
| Chpt 7, Sect 3
|
|
October 21 |
Multiplicative characters
| Chpt 8, Sect 1
|
|
October 23 |
Gauss Sums
| Chpt 8, Sect 2
|
|
October 25 |
Jacobi Sums
|
Chpt 8, Sect 3
|
October 28 |
More on Jacobi sums, Fermat equation
| Chpt 8, Sect 4, 5
|
|
October 30 |
More on Jacobi sums
| Chpt 8, Sect 6
|
|
Nov 1 |
Points on diagonal hypersurfaces
| Chpt 8, Sect 7
|
HW5 due Nov 11; Chpt 9, 19-25
|
Nov 4 |
Arithmetic of Eisenstein integers
| Chpt 9, Sect 1, 2
|
|
Nov 6 |
Cubic residues
| Chpt 9, Sect 3
|
|
Nov 8 |
Proof of cubic reciprocity
| Chpt 9, Sect 4
|
|
Nov 11 |
Intro to p-adic integers
| B&S: 1.3.1, 1.3.2
|
|
Nov 13 |
p-adic integers and p-adic numbers
| 1.3.2, 1.3.3
|
|
Nov 15 |
Convergence of p-adic numbers
| 1.3.4
|
|
Nov 18 |
The p-adics as a metric field
| 1.4
|
|
Nov 20 |
Congruences in the p-adics
| 1.5.1
|
HW 6 due Dec 4.
|
Nov 22 |
Hensel's Lemma
| 1.5.2
|
HW 6 due Dec 6: B&S p32, 2, 3, 4, 11; p40 1, 6; p 46, 1, 3 (fix the typo)
|