Math 3527 (Number Theory 1), Spring 2022
| Course Information | |||||
|---|---|---|---|---|---|
| Instructor | Class Times | Office Hours | |||
| Evan Dummit edummit at northeastern dot edu 571 Lake Hall |
(Sec 1) MWR 1:35pm-2:40pm, Snell Library 031 (Sec 2) MWR 10:30am-11:35am, 130 Dodge (note room change) |
MWR noon-1:00pm MWR 3:00pm-4:00pm Online via Zoom |
|||
| For detailed information about the course, please consult the 3527 Course Syllabus. (Note: any information given in class or on this webpage supersedes the written syllabus.) | |||||
| Teaching Assistant | Problem Session | ||||
| Mohamed Elbehiry (elbehiry.m at northeastern dot edu) | Friday, 3:00pm-4:30pm, online via Zoom | ||||
| Dmytro Matvieievskyi (matvieievskyi.d at northeastern dot edu) | Friday, noon-1:30pm, Nightingale 544 | ||||
| All homework assignments will be posted on this webpage (see below). | |||||
| The instructor will write lecture notes for the course (see below) in lieu of an official textbook as the semester progresses. To a moderate degree, the course will follow the presentation in J. Silverman's "A Friendly Introduction to Number Theory", but we will also add substantial additional material, and it will not be necessary to purchase the textbook for this course. | |||||
| The final exam is in Richards 458 from 3:30pm-5:30pm on Wed, May 4th. The instructor and TAs will hold various review sessions ahead of the exam. | |||||
| Homework Assignments | |||||||
|---|---|---|---|---|---|---|---|
| Some Tips on Problem Solving are available as suggestions for the written assignments. | |||||||
|
Homework #1, due Fri Jan 28th. (solutions)
Homework #2, due Fri Feb 4th. (solutions) Homework #3, due Fri Feb 11th. (solutions) Homework #4, due Fri Feb 18th. (solutions) Homework #5, due Fri Feb 25th. (solutions) Homework #6, due Fri Mar 11th. (solutions) (Mathematica code) Homework #7, due Fri Mar 25th. (solutions) (Mathematica code) Homework #8, due Fri Apr 1st. (solutions) (Mathematica code) Homework #9, due Fri Apr 8th. (solutions) Homework #10, due Fri Apr 22nd. (solutions) Homework #11, due Fri Apr 29th. (solutions) | |||||||
| Homework assignments are to be submitted 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 11pm eastern time. Late submissions, messy submissions, or otherwise unreadable submissions will be penalized at the grader's discretion. | |||||||
| Handouts / Lecture Notes | |||
|---|---|---|---|
| Handout | Topics | ||
| Chapter 1: The Integers (18pp, v3.00, posted 1/18) |
1.1 ~ The Integers, Axiomatically 1.2 ~ Divisibility and the Euclidean Algorithm 1.3 ~ Primes and Unique Factorization 1.4 ~ Rings and Other Number Systems |
||
| Chapter 2: Modular Arithmetic (20pp, v3.00, posted 1/26) | 2.1 ~ Modular Congruences and The Integers Modulo m 2.2 ~ Linear Equations Modulo m and The Chinese Remainder Theorem 2.3 ~ Powers Modulo m: Orders, Fermat's Little Theorem, Wilson's Theorem, Euler's Theorem 2.4 ~ Repeating Decimals |
||
| Chapter 3: Cryptography and Related Topics (26pp, v2.00, posted 2/14) | 3.1 ~ Overview of Cryptography 3.2 ~ Rabin Encryption 3.3 ~ RSA Encryption 3.4 ~ Zero-Knowledge Proofs 3.5 ~ Primality and Compositeness Testing 3.6 ~ Factorization Algorithms |
||
| Chapter 4: Unique Factorization and Applications (34pp, v3.00, posted 3/21) | 4.1 ~ Integral Domains, Euclidean Domains, and Unique Factorization 4.2 ~ Modular Arithmetic in Euclidean Domains 4.3 ~ Arithmetic in F[x] 4.4 ~ Arithmetic in Z[i] |
||
| Chapter 5: Squares and Quadratic Reciprocity (24pp, v3.00, posted 4/14) | 5.1 ~ Quadratic Residues and the Legendre Symbol 5.2 ~ The Law of Quadratic Reciprocity 5.3 ~ The Jacobi Symbol 5.4 ~ Applications of Quadratic Reciprocity 5.5 ~ Generalizations of Quadratic Reciprocity |
||
| Exam Information | |||
|---|---|---|---|
| Exam | Date, Time, Location | Topics | Review Material |
| Midterm 1 (Form A), (solutions) (Form B), (solutions) |
Wed, Mar 2nd In Class |
Homeworks 1-5 Notes 1.1-2.4 |
Review Problems (answers) In-class review Mon Feb 28th |
| Midterm 2 (Form A), (solutions) (Form B), (solutions) |
Wed, Apr 13th In Class |
Homeworks 6-9 Notes 3.1-4.3.2 |
Review Problems (answers) In-class review Mon Apr 11th |
| Final | Wed, May 4th 3:30pm-5:30pm, Richards 458 |
The final is COMPREHENSIVE! Homeworks 1-11 Notes Chapters 1-5 |
Part B Review Problems (answers) Review session date and time TBA |
| On all exams, calculators are permitted though generally they are not needed, and you are allowed a 1-page note sheet (8.5in by 11in, both sides) on which you may write or type anything. | |||
| 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 notes from someone who did attend. You are responsible for all material covered in lecture. | ||
| Read the Lecture Notes (or Textbook) | The lecture notes and the textbook are comprehensive sources 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 is a 3-hour weekly problem session run by the course TAs. This session runs from 9am-noon on Fridays in 544 Nightingale Hall, ahead of the time the homework is due. The goal of the problem session is 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. | ||
| Use Tutoring Services | Please consult the Northeastern Tutoring page for additional information on peer tutoring. Tutoring appointments can be made via MyNEU (which will provide lists of available appointments for the tutors for your specific classes); walk-in tutoring is very limited and tends to be unavailable near exam dates. | ||
| Course Schedule | |||
|---|---|---|---|
| The schedule is subject to change! All sections refer to the course lecture notes. | |||
| Week | Schedule | ||
| Week of Jan 17 (class starts 1/19) |
§1.1: The Integers, Axiomatically No homework this week. |
||
| Week of Jan 24 | §1.2: Divisibility and the Euclidean Algorithm §1.3: Primes and Unique Factorization §1.4: Rings and Other Number Systems Homework #1 due Friday, Jan 28th. |
||
| Week of Jan 31 | §1.4: Rings and Other Number Systems §2.1.1: Modular Congruences §2.1.2: Residue Classes §2.1.3: Modular Arithmetic §2.1.4: Units in Z/mZ Homework #2 due Friday, Feb 4th. |
||
| Week of Feb 7 |
§2.1.5: Zero Divisors in Z/mZ §2.2: Linear Equations Modulo m and The Chinese Remainder Theorem §2.3.1: Orders of Elements Modulo m §2.3.2: Fermat's Little Theorem, Wilson's Theorem Homework #3 due Friday, Feb 11th. |
||
| Week of Feb 14 |
§2.3.3: The Euler φ-function and Euler's Theorem §2.3.4: Primitive Roots and Discrete Logarithms §2.4: Repeating Decimals Homework #4 due Friday, Feb 18th. |
||
| Week of Feb 21 (no class 2/21) |
§3.1: Overview of Cryptography §3.2: Rabin Encryption Homework #5 due Friday, Feb 25th. |
||
| Week of Feb 28 | Review for Midterm 1. MIDTERM 1 in class on Wed, Mar 2nd §3.3: RSA Encryption No homework this week. |
||
| Week of Mar 7 | §3.4: Zero-Knowledge Proofs §3.5: Primality and Compositeness Testing §3.6: Factorization Algorithms Homework #6 due Friday, Mar 11th. |
||
| Spring Break (no classes) from Mar 14 to Mar 18 | |||
| Week of Mar 21 | §4.1.1: Norms and Z[√D] §4.1.2: Integral Domains and Common Divisors §4.1.3: Irreducible and Prime Elements §4.1.4: Euclidean Domains and Division Algorithms §4.1.5: Z[i] and F[x] as Euclidean Domains §4.1.6: Unique Factorization in Euclidean Domains Homework #7 due Friday, Mar 25th. |
||
| Week of Mar 28 | §4.2.1: Modular Congruences and Residue Classes §4.2.2: Arithmetic in R/rR §4.2.3: Units and Zero Divisors in R/rR §4.2.4: The Chinese Remainder Theorem §4.2.5: Orders, Euler's Theorem, Fermat's Little Theorem §4.3.1: Polynomial Functions, Roots of Polynomials Homework #8 due Friday, Apr 1st. |
||
| Week of Apr 4 | §4.3.2: Finite Fields §4.3.3: Primitive Roots §4.4.1: Residue Classes in Z[i] Homework #9 due Friday, Apr 8th. |
||
| Week of Apr 11 | Review for Midterm 2. MIDTERM 2 in class on Wed, Apr 13th §4.4.2: Factorization in Z[i] No homework this week. |
||
| Week of Apr 18 (no class 4/18) |
§5.1: Quadratic Residues and Legendre Symbols §5.2: The Law of Quadratic Reciprocity Homework #10 due Friday, Apr 22nd. |
||
| Week of Apr 25 (classes end 4/27) |
§5.3: Jacobi Symbols §5.4: Applications of Quadratic Reciprocity Review for Final Exam Homework #11 due Friday, Apr 29th. |
||
| FINAL EXAM will be held at from 3:30pm-5:30pm on Wednesday May 4th, Richards 458 | |||