Math 4527 (Number Theory 2), Fall 2022
| Course Information | |||||
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| Instructor | Class Times | Office Hours | |||
| Evan Dummit edummit at northeastern dot edu |
MWR 10:30am-11:35am Snell Library 043 |
MW noon-1:00pm MWR 3:00pm-4:00pm Online, via Zoom |
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| We will use Piazza for any course-related discussion: here is the Piazza page. | |||||
| For detailed information about the course, please consult the 4527 Course Syllabus. Note: any information given in class or on this webpage supersedes the written syllabus. | |||||
| All homework assignments will be posted on this webpage (see below). Homework assignments will be collected via Gradescope on Canvas. Please submit scans of your homework pages by 11:59pm Eastern on the due date. Late assignments may be penalized at the grader's discretion. |
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| The instructor will write lecture notes for the course (see below) in lieu of an official textbook as the semester progresses. The material we will cover in the course is drawn from a variety of sources, and will also depend somewhat on student interest; as such, it is difficult to give recommendations for a textbook. | |||||
| Homework Assignments + Exams | |||||||
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| Some Tips on Problem Solving are available as suggestions for the homework assignments. | |||||||
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Homework #1, due Fri Sep 16th. (solutions)
Homework #2, due Fri Sep 23rd. (solutions) Homework #3, due Fri Sep 30th. (solutions) Homework #4, due Fri Oct 7th. (solutions) Homework #5, due Fri Oct 14th. (solutions) Homework #6, due Fri Oct 21st. (solutions) Homework #7, due Sun Oct 30th. (solutions) Homework #8, due Sun Nov 6th. (solutions) Homework #9, due Sun Nov 13th. (solutions) Homework #10, due Sun Nov 20th. (solutions) Homework #11, due Fri Dec 2nd. Homework #12, due Fri Dec 9th. The final exam is optional and is only available for students who are below an A grade after Homework #12 in case they want to increase their grade. You will be notified via Canvas by Sunday Dec 11th whether you already have an A (and are thus exempt from the final exam) or whether you are below an A (and may thus wish to take the final exam to raise your grade). | |||||||
| Handouts / Lecture Notes | |||
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| Handout | Topics | ||
| Chapter 6: Rational Approximation and Diophantine Equations (35pp, v3.00, posted 9/5) | 6.1 ~ Some Diophantine Equations (Linear Diophantine Equations, the Frobenius Coin Problem, Pythagorean Triples) 6.2 ~ Rational Approximation and Transcendence (Farey Sequences, Finite and Infinite Continued Fractions, Rational Approximation, Irrationality and Transcendence) 6.3 ~ Pell's Equation (Motivation and Examples, General Structure, The Super Magic Box) 6.4 ~ An Assortment of Other Diophantine Equations |
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| Chapter 8: Quadratic Integer Rings (42pp, v3.00, posted 9/29) | 8.1 ~ Arithmetic in Rings and Domains (Ideals, Quotient Rings, Maximal and Prime Ideals, Integral Domains, Quadratic Integer Rings, Euclidean Domains, Principal Ideal Domains, Unique Factorization Domains, The Chinese Remainder Theorem) 8.2 ~ Factorization in Quadratic Integer Rings (Unique Factorization of Elements, Ideals in OD, Divisibility and Factorization of Ideals in OD, Calculating Factorizations) 8.3 ~ Applications of Factorization in Quadratic Integer Rings (Factoring in Z[i], O-2, O-3, Diophantine Equations, Cubic Reciprocity, Quartic Reciprocity) | ||
| Chapter 9: The Geometry of Numbers (27pp, v2.00, posted 10/22) | 9.1 ~ Minkowski's Convex-Body Theorem and Applications (Minkowski's Theorem, Sums of Two and Four Squares, Sums of Three Squares) 9.2 ~ Ideal Class Groups of Quadratic Integer Rings (The Ideal Class Group, Minkowski's Bound) 9.3 ~ Binary Quadratic Forms (Representation of Integers, Equivalence, Composition, Relation to Ideal Class Groups) |
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| Chapter 10: Analytic Number Theory (17pp, v1.00, updated 12/5) (updates in v1.00: added 10.2) |
10.1 ~ The Riemann Zeta Function and Dirichlet's Theorem on Primes in Arithmetic Progressions (The Riemann Zeta Function, Dirichlet Series, Group Characters, Dirichlet L-Series, Dirichlet's Theorem) 10.2 ~ Dedekind Zeta Functions and the Analytic Class Number Formula |
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| Tips For Success In This Course | |||
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| Attend Lecture | Missing lecture is a bad idea! If for any reason you cannot make it to a class, you should review the notes and watch the lecture recording. You are responsible for all material covered in lecture. | ||
| Read the Lecture Notes | The lecture notes are a comprehensive source of material for the course. The notes are intended as review material, although many students like to read them as preparation before attending the lecture on the corresponding topics. Please note that the electronic notes are not identical to the material covered in class: this is by design, so as to provide you a slightly different perspective on the material. | ||
| Solve Homework Problems | Much of the learning in this course will take place as you solve the homework problems. Like many other activities, problem-solving and proof-writing are things that are learned by doing them, not by hearing someone else tell you about them or reading about them in a book. As such, the homework assignments are an integral part of the course, and are fundamental to learning the material. It is highly recommended that you look over the homework assignments as soon as they are available, and work on them well in advance of the deadline: many problems will take substantial time and effort to solve, and you should expect to spend as much time as you need to finish the assignments. | ||
| Attend Office Hours | Office hours are specifically reserved for you to receive individual, one-on-one help from the instructor. Office hours will be the most effective when you have already put in effort to learn the material on your own (including trying to solve the homework problems), and when you come in with a list of specific questions or topics you are struggling with. | ||
| Course Schedule | |||
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| The schedule is subject to change! All sections refer to the course lecture notes. | |||
| Week | Schedule | ||
| Week of Sep 5 (class starts 9/7) |
§6.1.1: Linear Diophantine Equations §6.1.2: The Frobenius Coin Problem No homework this week. |
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| Week of Sep 12 | §6.1.3: Pythagorean Triples §6.2.1: Farey Sequences §6.2.2: Continued Fractions Homework #1 due Friday 9/16 on Canvas. |
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| Week of Sep 19 | §6.2.3: Infinite Continued Fractions §6.2.4: Rational Approximation §6.2.5: Irrationality and Transcendence §6.3.1: Pell's Equation Examples §6.3.2: Pell's Equation Properties Homework #2 due Friday 9/23 on Canvas. |
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| Week of Sep 26 |
§6.3.3: The Super Magic Box §6.4.1: Assorted Diophantine Equations §6.4.2: The Fermat Equation xn+yn=zn §8.1.1: Ideals of Commutative Rings §8.1.2: Quotient Rings Homework #3 due Friday 9/30 on Canvas. |
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| Week of Oct 3 | §8.1.3: Maximal and Prime Ideals §8.1.4: Arithmetic in Integral Domains §8.1.5: Quadratic Fields and Quadratic Integer Rings §8.1.6: Euclidean Domains §8.1.7: Principal Ideal Domains Homework #4 due Friday 10/7 on Canvas. |
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| Week of Oct 10 (no class 10/10) |
§8.1.8: Unique Factorization Domains §8.1.9: The Chinese Remainder Theorem §8.2.1: Unique Factorization of Elements in OD Homework #5 due Friday 10/14 on Canvas. |
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| Week of Oct 17 | §8.2.2: Ideals in OD §8.2.3: Divisibility and Unique Factorization of Ideals in OD §8.2.4: Calculating Factorizations in OD §8.3.1: Factorization in Z[i] Homework #6 due Friday 10/21 on Canvas. |
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| Week of Oct 24 | §8.3.2: Factorization in O√-2 and O√-3 §8.3.3: More Diophantine Equations §8.3.4: Cubic Reciprocity Homework #7 due Sunday 10/30 on Canvas. |
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| Week of Oct 31 | §8.3.4: Cubic Reciprocity §8.3.5: Quartic Reciprocity §9.1.1: Minkowski's Convex-Body Theorems §9.1.2: Sums of Two and Four Squares §9.1.3: Sums of Three Squares Homework #8 due Sunday 11/6 on Canvas. |
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| Week of Nov 7 | §9.2.1: The Ideal Class Group §9.2.2: Minkowski's Bound §9.3.1: Representations by Quadratic Forms Homework #9 due Sunday 10/13 on Canvas. |
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| Week of Nov 14 | §9.3.2: Equivalence of Quadratic Forms §9.3.3: Composition of Quadratic Forms §9.3.4: Dirichlet's Composition Law Homework #10 due Sunday 10/20 on Canvas. |
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| Week of Nov 21 (no class 11/23-11/27) | §9.3.5: Quadratic Forms and Ideal Class Groups No homework this week. |
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| Week of Nov 28 | §10.1.1: The Riemann Zeta Function §10.1.2: Motivation for Dirichlet's Theorem §10.1.3: Dirichlet Series §10.1.4: Group Characters and Dirichlet Characters Homework #11 due Friday 12/2 on Canvas. |
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| Week of Dec 5 (class ends 12/7) |
§10.1.5: Dirichlet L-Series and Dirichlet's Theorem §10.2.1: Dedekind Zeta Functions §10.2.2: The Analytic Class Number Formula Homework #12 due Friday 12/9 on Canvas. |
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| Optional Final Exam due 12/15 | |||