Math 3527 (Number Theory 1, Course #35849/35248), Spring 2020
| Course Information | |||||
|---|---|---|---|---|---|
| Instructor | Class Times | Office Hours | |||
| Evan Dummit edummit at northeastern dot edu |
Effective Thursday, March 12th, all Northeastern in-person meetings are cancelled until further notice. | ||||
| In lieu of office hours, 3527 now has a Piazza page. Links to all of the live lecture material are hosted there. | |||||
| For detailed information about the course, please consult the 3527 Course Syllabus (Sec 01) (Sec 02). (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 are now to be submitted via the 3527 Blackboard 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 try to submit the pages in order. Assignments are due at 6pm eastern time. |
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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. 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. | |||||
| Homework Assignments | |||||||
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| Some Tips on Problem Solving are available as suggestions for the written assignments. | |||||||
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Homework #1, due Wed Jan 15th. (solutions)
Homework #2, due Thu Jan 23nd. (solutions) Homework #3, due Wed Jan 29th. (solutions) Homework #4, due Wed Feb 5th. (solutions) Homework #5, due Wed Feb 12th. (solutions) Homework #6, due Wed Feb 19th. (Sample Mathematica code.) (solutions) Homework #7, due Wed Mar 11th. (Sample Mathematica code.) (solutions) Homework #8, due Sat Mar 21st. (solutions) Homework #9, due Fri Mar 27th. (solutions) Homework #10, due Fri Apr 3rd. (solutions) Homework #11, due Fri Apr 17th. (solutions) | |||||||
| Lecture Slides | |||
|---|---|---|---|
| These are slides corresponding to the material that would have been covered in the lectures. | |||
| Date | Material | ||
| Thu, Mar 12th | Lecture 23: The Chinese Remainder Theorem, Fermat's Little Theorem, Euler's Theorem (Notes 4.2.4 + 4.2.5) | ||
| Wed, Mar 18th | Lecture 24: Polynomials as Functions, Factorization in F[x] (Notes 4.3.1) | ||
| Thu, Mar 19th | Lecture 25 Finite Fields (Notes 4.3.2) | ||
| Mon, Mar 23rd | Lecture 26: Primitive Roots (Notes 4.3.3) | ||
| Wed, Mar 25th | Lecture 27: Modular Arithmetic in Z[i] (Notes 4.4.1) | ||
| Thu, Mar 26th | Lecture 28: Factorization in Z[i] (Notes 4.4.2) | ||
| Week of Mar 30 - Apr 3 | No new content, review of past material + midterm 2 preparation | ||
| Mon, Apr 6th | Lecture 29: Polynomial Congruences (Notes 5.1) | ||
| Wed, Apr 8th | Lecture 30: Quadratic Residues + Legendre Symbols (Notes 5.2) | ||
| Thu, Apr 9th | Lecture 31: Quadratic Reciprocity + Jacobi Symbols (Notes 5.3 + 5.4) | ||
| Mon, Apr 13th | Lecture 32: Applications of Quadratic Reciprocity (Notes 5.5) | ||
| Handouts / Lecture Notes | |||
|---|---|---|---|
| Handout | Topics | ||
| Chapter 1: The Integers (17pp, v2.05, updated 2/23) (updates in 2.05: added definition of F[x] to 1.4.1, changed example at end of 1.4.2 to discuss non-unique factorization) |
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 |
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| Chapter 2: Modular Arithmetic (20pp, v2.00, posted 1/12) | 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 |
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| Chapter 3: Cryptography and Related Topics (29pp, v2.00, posted 2/4) | 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 |
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| Chapter 4: Unique Factorization and Applications (32pp, v2.10, updated 3/13) (updates in 2.10: fixed typos and clarified explanations, removed duplicate example in 4.2.3, added example of Euler in 4.2.5, added example at end of 4.3.3, fixed example at end of 4.4.1, added missing proposition to start of 4.4.2) |
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] |
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| Chapter 5: Squares and Quadratic Reciprocity (27pp, v2.00, posted 4/6) | 5.1 ~ Polynomial Congruences and Hensel's Lemma 5.2 ~ Quadratic Residues and the Legendre Symbol 5.3 ~ The Law of Quadratic Reciprocity 5.4 ~ The Jacobi Symbol 5.5 ~ Applications of Quadratic Reciprocity 5.6 ~ Generalizations of Quadratic Reciprocity |
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| Exam Information | |||
|---|---|---|---|
| Exam | Date, Time, Location | Topics | Review Material |
| Midterm 1 (Form A), (solutions) (Form B), (solutions) |
Thu, February 20th In Class |
Homeworks 1-6 Notes Chapters 1-3 Topics List |
Review Problems Review Answers In-class review Wed Feb 19th |
| Midterm 2 (Form A), (solutions) (Form B), (solutions) |
Sun, Apr 5th or alternate date by arrangement |
Homeworks 7-10 Notes Chapter 4 Topics List |
Review Problems Review Answers Review session on Thu Apr 2nd |
| Final | Tue, Apr 21st or alternate date by arrangement |
The final is COMPREHENSIVE! Homeworks 1-11 Notes Chapters 1-5 (skip 5.3.2, 5.6) Full Topics List |
Review session on Sat Apr 18th or Sun Apr 19th |
| Bring your University ID to all exams. Cell phones, electronic or calculating devices, books, and notes of any kind will NOT be permitted in exams. | |||
| 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 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. | ||
| Attend Problem Sessions | There is a 3-hour weekly problem session run by the course's TA. This session runs from 9am-noon on Tuesdays in 544 Nightingale Hall. 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 assistance from the TA. | ||
| Use Tutoring Services | The university offers a wide variety of (free) tutoring services. Please consult the Northeastern Tutoring page for additional information on peer tutoring. The Math Tutoring Center, located in 540B NI, is specifically set up for tutoring in mathematics courses. 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 | |||
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| The schedule is subject to change! All sections refer to the course lecture notes. | |||
| Week | Schedule | ||
| Week of Jan 6 (class starts 1/6) |
§1.1: The Integers, Axiomatically §1.2: Divisibility and the Euclidean Algorithm No homework this week. |
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| Week of Jan 13 | §1.3: Primes and Unique Factorization §1.4: Rings and Other Number Systems §2.1.1: Modular Congruences §2.1.2: Residue Classes Homework #1 due Wednesday, Jan 15th. |
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| Week of Jan 20 (no class 1/20) |
§2.1.3: Modular Arithmetic §2.1.4: Units in Z/mZ §2.1.5: Zero Divisors in Z/mZ §2.2: Linear Equations Modulo m and The Chinese Remainder Theorem Homework #2 due Thursday, Jan 23rd. |
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| Week of Jan 27 |
§2.3.1: Orders of Elements Modulo m §2.3.2: Fermat's Little Theorem, Wilson's Theorem §2.3.3: The Euler φ-function and Euler's Theorem Homework #3 due Wednesday, Jan 29th. |
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| Week of Feb 3 |
§2.3.4: Primitive Roots and Discrete Logarithms §2.4: Repeating Decimals §3.1: Overview of Cryptography §3.2: Rabin Encryption §3.3: RSA Encryption Homework #4 due Wednesday, Feb 5th. |
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| Week of Feb 10 | §3.3: RSA Encryption §3.4: Zero-Knowledge Proofs §3.5: Primality and Compositeness Testing §3.6: Factorization Algorithms Homework #5 due Wednesday, Feb 12th. |
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| Week of Feb 17 (no class 2/17) |
Review for Midterm 1. Homework #6 due Wednesday, Feb 19th. |
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| MIDTERM 1 will be held in class on Thursday, February 20th | |||
| Week of Feb 24 | §3.6: Factorization Algorithms §4.1.1: Integral Domains §4.1.2: Euclidean Domains and Division Algorithms §4.1.3: Irreducible and Prime Elements No homework this week. |
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| Spring Break (no classes) from Feb 29 to Mar 8 | |||
| Week of Mar 9 | §4.1.4: Unique Factorization Domains §4.2: Modular Arithmetic in Euclidean Domains Homework #7 due Wednesday, Mar 11th. |
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| Effective Thursday, March 12th, all in-person meetings are cancelled. | |||
| Week of Mar 16 | §4.3.1: Polynomial Functions, Roots of Polynomials §4.3.2: Finite Fields Homework #8 due Saturday, Mar 21st. |
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| Week of Mar 23 | §4.3.3: Primitive Roots §4.4.1: Residue Classes in Z[i] §4.4.2: Factorization in Z[i] Homework #9 due Friday, Mar 27th. |
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| Week of Mar 30 | Review for Midterm 2. Homework #10 due Friday, Apr 3rd. |
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| MIDTERM 2 will be held on Sunday, April 5th (alternate dates of Sat Apr 4th and Mon Apr 6th are available) | |||
| Week of Apr 6 | §5.1: Polynomial Congruences and Hensel's Lemma §5.2: Quadratic Residues and Legendre Symbols §5.3.1: Motivation for Quadratic Reciprocity §5.4: Jacobi Symbols No homework this week. |
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| Week of Apr 13 (classes end 4/14) |
§5.5: Applications of Quadratic Reciprocity Review for Final Exam Homework #11 due Friday, Apr 17th. |
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| FINAL EXAM will be held on Tuesday, April 21st (alternate dates of Mon Apr 20th and Wed Apr 22nd are available) | |||