Math 4527 (Number Theory 2), Spring 2024
| Course Information | |||||
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| Instructor | Class Times | Office Hours | |||
| Evan Dummit edummit at northeastern dot edu 571 Lake Hall |
MWR 4:40pm-5:45pm Shillman 325 |
R 12:10pm-1:10pm MR 3:05pm-4:05pm Online via Zoom |
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| 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.) | |||||
| We will use Piazza for any course-related discussion: here is the 4527 Piazza page. | |||||
| Teaching Assistant | Problem Session | ||||
| Ryan Kannanaikal | Fridays 11am-1pm, Richards 160 | ||||
| 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 Jan 19th. (solutions)
Homework #2, due Fri Jan 26th. (solutions) Homework #3, due Fri Feb 2nd. (solutions) Homework #4, due Fri Feb 9th. (solutions) Homework #5, due Fri Feb 16th. (solutions) Homework #6, due Fri Feb 23rd. (solutions) Homework #7, due Fri Mar 1st. (solutions) Homework #8, due Fri Mar 15th. (solutions) Homework #9, due Fri Mar 22nd. (solutions) Homework #10, due Fri Mar 29th. (solutions) Homework #11, due Fri Apr 5th. (solutions) Homework #12, due Fri Apr 12th. Final Exam and the Short Final Exam due Fri Apr 26th. It is listed on Canvas whether you should take the regular final or the short final. | |||||||
| Homework assignments are to be submitted to Gradescope via the course's Canvas page. To submit an assignment, navigate to "Assignments" and select the appropriate homework assignment. Then attach scans of each page of your assignment (or a pdf) and click Submit. Please submit the pages in order and verify that all pages are included and uploaded correctly. You may resubmit as many times as you like. Assignments are due at 11:59pm eastern time. Late submissions, messy submissions, or otherwise unreadable submissions will be penalized at the grader's discretion. Ensure you mark all problem pages when submitting to Gradescope; failure to do so may result in point penalties. | |||||||
| Handouts / Lecture Notes | |||
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| Handout | Topics | ||
| Chapter 6: Rational Approximation and Diophantine Equations (35pp, v3.00, posted 1/7) | 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, v3.50, posted 1/30) | 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, v3.00, posted 3/9) | 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, v1.50, posted 3/27) | 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 | |||
|---|---|---|---|
| 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 Problem Sessions | There are weekly problem sessions run by the course TAs. The goal of the problem sessions are to provide you a location where you can work collaboratively with other students on assignments, and also get TA assistance. | ||
| 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 Jan 8 (class starts 1/8) |
§6.1.1: Linear Diophantine Equations §6.1.2: The Frobenius Coin Problem §6.1.3: Pythagorean Triples §6.2.1: Farey Sequences No homework this week. |
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| Week of Jan 15 (no class 1/15) |
§6.2.2: Continued Fractions §6.2.3: Infinite Continued Fractions §6.2.4: Rational Approximation Homework #1 due Friday, Jan 19th. |
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| Week of Jan 22 | §6.2.5: Irrationality and Transcendence §6.3.1: Pell's Equation Examples §6.3.2: Pell's Equation Properties §6.3.3: The Super Magic Box Homework #2 due Friday, Jan 26th. |
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| Week of Jan 29 |
§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, Feb 2nd. |
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| Week of Feb 5 |
§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 Homework #4 due Friday, Feb 9th. |
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| Week of Feb 12 | §8.1.7: Principal Ideal Domains §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, Feb 16th. |
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| Week of Feb 19 (no class 2/19) |
§8.2.2: Ideals in OD §8.2.3: Divisibility and Unique Factorization of Ideals in OD §8.3.2: Factorization in O√-2 and O√-3 Homework #6 due Friday, Feb 23rd. |
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| Week of Feb 26 | §8.3.3: More Diophantine Equations §8.3.4: Cubic Reciprocity §8.3.5: Quartic Reciprocity Homework #7 due Friday, Mar 1st. |
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| Spring Break (no classes) from Mar 4 to Mar 8 | |||
| Week of Mar 11 | §9.1.1: Minkowski's Convex-Body Theorems §9.1.2: Sums of Two and Four Squares §9.1.3: Sums of Three Squares §9.2.1: The Ideal Class Group Homework #8 due Friday, Mar 15th. |
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| Week of Mar 18 | §9.2.2: Minkowski's Bound §9.3.1: Representations by Quadratic Forms §9.3.2: Equivalence of Quadratic Forms Homework #9 due Friday, Mar 22nd. |
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| Week of Mar 25 | §9.3.3: Composition of Quadratic Forms §9.3.4: Dirichlet's Composition Law §9.3.5: Quadratic Forms and Ideal Class Groups Homework #10 due Friday, Mar 29th. |
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| Week of Apr 1 | §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, Apr 5th. |
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| Week of Apr 8 (no class 4/8) |
§10.1.5: Dirichlet L-Series and Dirichlet's Theorem §10.2.1: Dedekind Zeta Functions Homework #12 due Friday, Apr 12th. |
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| Week of Apr 15 (no class 4/15, classes end 4/17) |
§10.2.2: The Analytic Class Number Formula No homework this week. |
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| The takehome final exam is due on Thursday, April 24th. | |||