| General Information | Schedule | Homework |
| Schedule |
|---|
We will cover most of the material in Bolker, along with the first several chapters of Stewart/Tall. However, we will not follow them particularly closely. You should read the material in the corresponding sections to supplement the course notes.
| Class Number | Date | Reading | Brief Description |
|---|---|---|---|
| 1 | Monday, 1/25 | Bolker: Section 1 ST: Introduction, 1.1 |
Introduction, Sums of Squares, Review of Rings, Ideals |
| 2 | Wednesday, 1/27 | Bolker: Sections 1, 2 ST: Introduction, 1.1 |
Divisibility, GCDs, Primes |
| 3 | Friday, 1/29 | Bolker: Sections 2, 5 ST: Introduction, 1.1 |
The Fundamental Theorem of Arithmetic, Consequences |
| 4 | Monday, 2/1 | Bolker: Sections 3, 4, 7 | Linear Equations over Z, Congruences, The Ring Z/nZ |
| 5 | Wednesday, 2/3 | Bolker: Sections 7, 9 | Solving Linear Congruences, Euler's Theorem, Fermat's Little Theorem |
| 6 | Friday, 2/5 | Bolker: Sections 8, 10 | Chinese Remainder Theorem, Properties of the Euler Phi-Function |
| 7 | Monday, 2/8 | Bolker: Section 13 | Wilson's Theorem, Square Roots of -1 in Z/pZ |
| 8 | Wednesday, 2/10 | Bolker: Section 17 | Characterizing Cyclic Groups, Understanding U(Z/nZ) |
| 9 | Friday, 2/12 | Bolker: Sections 18, 19 | Primitive Roots, Classifying U(Z/pZ) and U(Z/2^kZ) |
| 10 | Monday, 2/15 | Bolker: Sections 19, 20 | Classifying U(Z/2^kZ) and U(Z/p^2Z) |
| 11 | Wednesday, 2/17 | Bolker: Sections 20, 21 | Primitive Roots Modulo Odd Prime Powers, Classifying U(Z/nZ) |
| 12 | Friday, 2/19 | Bolker: Sections 21, 23 | Power Residues |
| 13 | Monday, 2/22 | Bolker: Section 23 | Quadratic Residues, The Legendre Symbol |
| 14 | Wednesday, 2/24 | Bolker: Section 24 | Gauss' Lemma and Consequences |
| 15 | Friday, 2/26 | Bolker: Section 25 | Quadratic Reciprocity |
| 16 | Monday, 3/1 | - | Primes, Irreducibles, PIDs |
| 17 | Wednesday, 3/3 | - | Prime and Maximal Ideals, Noetherian Rings, UFDs |
| 18 | Friday, 3/5 | - | Euclidean Domains, The Gaussian Integers |
| 19 | Monday, 3/8 | - | Factorizations in the Gaussian Integers, Sums of Squares |
| 20 | Wednesday, 3/10 | - | Sums of Squares, Primes in the Gaussian Integers |
| 21 | Friday, 3/12 | Bolker: Section 28 | Field Extensions, Algebraic Elements, Minimal Polynomials |
| 22 | Monday, 3/15 | Bolker Section 28 ST: Section 1.2 |
Extensions by an Algebraic Element |
| 23 | Wednesday, 3/17 | ST: Sections 1.2, 1.3 | Extensions by an Algebraic Element, Gauss' Lemma for Polynomials, Irreducibility Criteria |
| 24 | Friday, 3/19 | ST: Section 1.3 | Eisenstein's Criterion, Finite and Algebraic Extensions, Number Fields |
| - | Monday, 3/22 | - | Spring Break |
| - | Wednesday, 3/24 | - | Spring Break |
| - | Friday, 3/26 | - | Spring Break |
| - | Monday, 3/29 | - | Spring Break |
| - | Wednesday, 3/31 | - | Spring Break |
| - | Friday, 4/2 | - | Spring Break |
| 25 | Monday, 4/5 | Bolker: Sections 28, 29 | Algebraic Integers |
| 26 | Wednesday, 4/7 | Bolker: Sections 28, 29 | Minimal Polynomials, Algebraic Integers in Q(i) |
| 27 | Friday, 4/9 | Bolker: Section 29 ST: Section 3.1 |
Quadratic Number Fields |
| 28 | Monday, 4/12 | Bolker: Sections 29, 30 ST: Section 3.1 |
Algebraic Integers in Quadratic Number Fields |
| 29 | Wednesday, 4/14 | Bolker: Sections 30, 31 | Norms and Units in Quadratic Number Fields, Pell's Equation |
| 30 | Friday, 4/16 | Bolker: Section 31 | Rational Approximations, Pell's Equation |
| 31 | Monday, 4/19 | Bolker: Section 31 | The Unit Group in Real Quadratic Number Fields |
| 32 | Wednesday, 4/21 | Bolker: Sections 32, 33 ST: Chapter 4 |
Failures of Unique Factorization, Euclidean Quadratic Number Fields |
| 33 | Friday, 4/23 | Bolker: Section 33 ST: Chapter 4 |
Euclidean Quadratic Number Fields, Solving Diophantine Equations |
| 34 | Monday, 4/26 | ST: Chapter 4 | Solving Diophantine Equations |
| 35 | Wednesday, 4/28 | ST: Section 2.1 | Conjugates, Number Fields as Extensions by One Algebraic Element |
| 36 | Friday, 4/30 | ST: Section 2.2 | Embeddings of Number Fields |
| 37 | Monday, 5/3 | ST: Section 1.4 | Symmetric Polynomials |
| 38 | Wednesday, 5/5 | ST: Section 2.2 | Field Polynomials, Examples |
| 39 | Friday, 5/7 | ST: Sections 2.2, 2.5 | Field Polynomials as Powers of Minimal Polynomials, Norms and Traces |
| 40 | Monday, 5/10 | ST: Section 2.3 | Algebraic Integers in General Number Fields |
| 41 | Wednesday, 5/12 | ST: Section 2.4, 5.1, 5.2 | Structure of the Ring of Integers in a Number Field |
| 42 | Friday, 5/14 | - | Structure of the Ring of Integers in a Number Field |