Math 311: Number Theory

Useful Information:

Class: Tuesday, Thursday 9:50-11:10 in Chemistry 128
My office phone: (631) 632-8276
My Email: deland@math.sunysb.edu
My Office hours: Math Building 4-114, Monday 11:30-12:30, 4:30-5:30 Wednesday 11-12
There will be weekly assignments, one in-class midterm, and a final exam.
Midterm Date: Thursday, March 18th, in class.
The class reference will be the book of Niven, Zuckerman and Montgomery, An Introduction to the Theory of Numbers (5-th Edition). I will abbreviate this ITN below. We will mostly follow the text, though sometimes I will present results in a slightly different way or different order.

Course Outline:

The first half of the course will focus on Chapters two and three. The highlight will be the quadratic reciprocity theorem (first proved by Gauss). The material in Chapter 1 should mostly be familiar to you, I recommend rereading it before the course begins. We will cover it very briefly. In addition, I expect you to be familiar with basic group theory and ring theory. Our book introduces these concepts only at the end of Chapter 2. I view this as the book's main shortcoming, as many ideas and results are better expressed in this language. I have written some notes below - I highly recommend reading these before the course begins and referring back to them as necessary.

The material which we will cover in the second half of the course has not yet been decided on exactly. I am leaning toward talking about Diophantine Equations and/or Dirichlet's Theorem. Interest from the class will be taken into consideration!

Material Covered, Homework Assignments, and Lecture Notes

Notes on Groups, Rings, and Ideals. These notes are a work in progress, they will be updated as we (you and I) see necessary.
Homework 1. Due Date: Tuesday February 2nd
Tuesday January 26th: Discussed divisibility, the division algorithm, and gcds.
Read notes on Groups, Rings, Ideals. Read Chapter 1 of ITN.
Thursday January 28th: More on gcds, Prime numbers, unique factorization, binomial theorem.
Finish Reading Chapter 1.
Tuesday February 2nd: Congruence and solving congruence equations. Read Chapter 2.1 and 2.2.
Homework 2. Due Date: Tuesday February 9th.
Thursday February 4th: Discuss complete and reduced residue systems. Groups of units. Wilson, Euler, and Fermat Theorems. Start discussion about integers represented by sums of two squares.
Tuesday February 9th: Integers represented by sums of two squares. Chinese Remainder Theorem. Read Chapter 2.3.
Homework 3. Due Date: Tuesday February 16th.
Thursday February 11th: Class has been canceled by the University.
Homework 4. Due Date: Tuesday March 2nd. Note the extended due date.
Tuesday February 16th: Finish discussing the Chinese Remainder Theorem and prove Hensel's Lemma. Chapter 2.3, 2.6.
Thursday February 18th: Finish the discussion of Hensel's Lemma and begin talking about the little that can be said about solving polynomial congruences mod primes. Chapter 2.6 and 2.7.
Tuesday February 23rd: Class was cancelled.
Thursday February 25th: Finish the discussion of solving congruences mod primes. Begin talking about the structure of the group of units modulo some number m. Define primitive roots. Chapter 2.7 and 2.8.
Tuesday March 2nd: We discussed the existence of primitive roots mod powers of odd primes and how this can help us decide whether or not there are nth roots of a number a modulo the prime. Chapter 2.8.
Homework 5. Due Date: Tuesday March 9th.
Thursday March 4th: Which numbers have primitive roots? How can we use this to "take n-th" roots of numbers modulo those powers. Begin discussion of quadratic reciprocity. Chapter 2.8 and 3.1.
Homework 6. Due Date: Tuesday March 16th or March 23rd.
Midterm Review
Tuesday March 9th: A proof of the law of quadratic reciprocity. Chapter 3.1-2.
Thursday March 11th: Soliving linear Diophantine equations. Chapter 5.1-2.
Tuesday March 16th: Reviewed for Midterm.
Thursday March 18th: Midterm.
Tuesday March 23rd: Finished the discussion of linear equations and began talking about quadratic ones. Chapter 3.4.
Homework 7. Due Date: Tuesday April 6th
Thursday March 25th: Talked about Pythagorean triples and then Fermat's last theorem when n = 4.
Tuesday April 6th: Defined an equivalence relation on quadratic forms and talked about how to find a reduced member in any equivalence class. Chapter 3.5
Homework 8. Due Date: Tuesday April 13th
Thursday April 8th: Proved a formula on the number of ways to write a given integer as the sum of two squares. Chapter 3.6.
Tuesday April 13th: Discussed quadratic forms in 3 variables and proved a criterion for whether or not they admit solutions.
Homework 9. Due Date: Tuesday April 20th
Sum of Two Squres. An example computation showing how the theorem we proved in class actually lets us compute the ways to write a given number as the sum of two squares (if possible).
Thursday April 15th: Example day. We showed how to determine (in practice) if a quadratic form in three variables has a solution. We then discussed how to determine all possible ways of writing 8450 as the sum of two squares.
Monday April 19th - There was a typo in the Hint to Problem 5 on Homework 9. It has now been fixed!
Tuesday April 20th: Showed every integer is the sum of 4 squares. Proved that every positive definite, symmetric, integer (3 by 3) matrix of determinant 1 is equivalent to the identity matrix.
Thursday April 22nd: Mostly proved the theorem that integers which are not 4^a(8b + 7) are the sum of three squares. Discussed how to parameterize solutions to degree 2 equations given a single solution.
Homework 10. Due Date: Thursday April 29th - NOTE the Thursday due date.
Sum of Four Squares. We proved in class that every integer can be written as the sum of four squares. These notes reprove that theorem.
Sum of Three Squares. We mostly showed in class that every integer not of the form 4^a(8b + 7) can be written as the sum of three squares. These notes are supposed to remind you what it was that we did in class.
Tuesday April 27th: Introduced the rings Z[i] and Z[w] and proved some basic properties about them.
Thursday April 29th: Studied primes in the rings Z[i] and Z[w].
Algebraic Numbers. These are notes for the material covered in the last section of our class. I have rearranged the Chapter 9 of out book because I did not want to discuss the material in the same amount of depth that they do. Hopefully you will find this useful.
Homework 11. Due Date: May 6th - NOTE the Thursday due date. Also this is the last day of class.
Final Exam Review
Degree 3 Fermat Equation. These are notes for the material covered in the last lecture. We prove that the equation x^3 + y^3 = z^3 has no nontrivial solutions by working in the ring Z[w].
Extra Problems.. Some more problems concerning the material we've covered in the last two weeks.

Homework Hints and Solutions:

These are not intended to be complete solutions to the homework problems. It is possible that there are small typos, for which I apologize in advance. If you can not figure out what I have written or would like more explanation or would like other problems addressed - PLEASE come to my office hours or otherwise make an appointment.

Class and University Policies

Grading
The grading breakdown to compute your final score will be roughly 35% for the homework, 30% for the midterm, and 35% for the final exam.
Homework Return Policy
Graded problem sets and exams will be handed back in lecture. If you cannot attend the lecture in which a problem set or exam is handed back, it is your responsibility to contact your instructor and arrange a time to pick up the work (typically in office hours).

You are responsible for collecting any graded work by the end of the semester. After the end of the semester, the instructor is no longer responsible for returning your graded work.

Disability Support Services
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