Michael MOVSHEV 

MAT 311 
Number Theory

Fall 2021

We will meet on MW 2:40pm-4:00pm in Mathematics 4130

First day of class: Monday August 23, 2021.
Final exam : Final Exam: Wednesday, Dec. 8, 5:30pm-8:00pm

Office hours:
TBA.

How to contact me?
the best way is to email me there: mmovshev at math dot sunysb dot edu

The grader Daniel Brogan has office hours:F 10:00am-11:00am, W 9:00am-10:00am in MLC; M 10:00am-11:00am in his office Math Tower S-235M

Our textbook:
An Introduction to the Theory of Numbers (Hardcover), Wiley, Fifth edition (January 1991), by Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery

Link to Current Homework: The Homework is an important part of this class. I will take it from the book or from other sources. Click here to go to the homework page.

Course notes and announcements:

Quick intro: Number theory is certainly one of the oldest subject within mathematics. Already 36 centuries ago in tablets written in Babylone, there were examples of such problems. Some mathematicians like to say that it occupies within mathematics the same place as mathematics within science...Some people like to see it as the purest domain in mathematics, and yet some others like to see all its applications to cryptography, computer science,etc...
Number theory has the remarkable advantage of being able to formulate extremely deep problems almost without prerequisites. A model for this is certainly Fermat's last theorem, that can be stated in one line but that resisted all the efforts of mathematicians for centuries... For this reason, I think it is an excellent "entry point" to mathematics: we will start with very simple material like divisibility properties, congruences, continue with simple Diophantine equations, and slowly progress towards deeper questions like Quadratic reciprocity.
I will not hesitate to provide introductions to much recent material, like one and two-dimensional representations, or even the Absolute Galois group, which is nowadays one of the most mysterious objects of contemporary mathematics, and one that is certainly the center of a tremendous mathematical activity.

Prerequisites:
For this class you need to have taken MAT 312 or 313 or 318.

Link to Current Homework: Regularly you will have to consult this homework page to know what has been assigned.

Syllabus :

Day of

Sections Covered

Unit 1

Divisibility and Congruences

Week 1:Aug. 23

Section 1.1 Introduction, Section 1.2 Divisibility

Week 1:Aug. 25

Section 1.3 Primes, Section 1.4 The Binomial Theorem

Week 2:Aug. 30

Section 2.1 Congruences, Section 2.2 Solutions of Congruences

Week 2: Sept 1

Section 2.3 The Chinese Remainder Theorem

Week 3:Sept. 6

Labor Day- Classes not in Session

Week 3:Sept. 8

Section 2.6 Prime Power Moduli

Week 4:Sept. 13

Section 2.7 Prime Modulus

Week 4:Sept. 15

Section 2.8 Primitive Roots and Power Residues

Week 5:Sept. 20

Section 3.1 Quadratic Residues

Unit 2

Quadratic Reciprocity and Diophantine Equations

Week 5:Sept. 22

Section 3.2 Quadratic Reciprocity

Week 6:Sept. 27

Section 5.1 Binary Linear Forms, Section 5.2 Simultaneous Linear Equations

Week 6:Sept. 29

Midterm 1 on Unit 1

Week 7:Oct 4

Section 3.4 Binary Quadratic Forms

Week 7:Oct 6

Section 5.3 Pythagorean Triples

Week 8:Oct 11

Oct 11 Fall Break

Week 8:Oct 13

Section 3.5 Equivalence and Reduction of Binary Quadratic Forms

Week 9:Oct 18

Section 3.6 Sums of Two Squares

Week 9:Oct 20

Section 5.5 Ternary Quadratic Forms

Week 10:Oct 25

Midterm 2 on Unit 2

Unit 3

Special Cases of Fermat's Last Theorem

Week 10:Oct 27

Section 9.1 Polynomials, Section 9.2 Algebraic Numbers

Week 11:Nov 1

Section 9.3 Algebraic Number Fields, Section 9.4 Algebraic Integers

Week 11:Nov 3

Section 9.5 Quadratic Fields, Section 9.6 Units in Quadratic Fields

Week 12:Nov 8

Section 9.7 Primes in Quadratic Fields, Section 9.8 Unique Factorization, Section 9.9 Primes in Quadratic Fields with the UFD property

Week 12:Nov 10

Section 9.10 The Cubic Case of Fermat's Last Theorem

Week 13:Nov 15

Selected Topics from Chapter 7

Week 13:Nov 17

Selected Topics from Chapter 7

Week 14:Nov 22

Nov 24 Thanksgiving Break

Week 14:Nov 24

Selected Topics from Chapter 7

Week 15:Nov 29

Miscellaneous

Week 15:Dec 1

Miscellaneous

Week 16:Dec 6

Miscellaneous

Exams:

Midterm I

09/29/21

Usual room

Midterm II

10/25/21

Usual room  

Final

12/08/21

TBA

Homework and grading policy: Here is how your final grade will be computed. of the following:

Exam I

25%

Exam II

25% 

Final Exam

35%

Homework

15%



Late homework will not be accepted.

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