Michael MOVSHEV 

MAT 311 
Number Theory 

We will meet on MWF 10:40am-11:35am in Physics P123

First day of class: Monday January 28, 2008.
Final exam : TBA.

Office hours:
TBA.

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

The grader Caner Koca has office hours:Tu-Th: 11:30-12:30 in MLC Mo: 1-2 in his office (3-118)

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

Week 1:Jan. 28 ,30,Feb. 1

Divisibility, prime numbers,repartition of primes, rational points on circle

Week 2:Feb. 4,6,8

Congruences, Euclid algorithm, linear equations

Week 3:Feb. 11,13,15

Euler's phi function, summary about groups,rings,Chinese remainder theorem

Week 4:Feb. 18,20,22

Structure of the multiplicative group, existence of square roots

Week 5:Feb. 25,27,29

Review, Midterm 1

Week 6:March 3,5,7

Quadratic reciprocity theorem

Week 7:March 10,12,14

The RSA cryptosystem, Rabin's system, basic attacks on RSA

Week 8:March 17,19,21

Spring Recess

Week 9:March 24,26,28

Ideals, quotient of a ring by an ideal, quadratic extensions

Week 10:March 31, Apr 2,4

Prime ideals (continued), basic intro to topological spaces,Spec of a ring Review, Midterm II

Week 11:April 7 ,9,11

Continued fractions and approximations of real numbers

Week 12:April 14,16,18

Intro to elliptic functions and cryptography

Exams:

Midterm I 

Wed. Feb 27th

Usual room

Midterm II  

Fr. Apr 4th

Usual room  

Final  

TBA

Usual room

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%
Your intermediate Spring Term grade can be found here. The intermediate grade are based on your homework performance and the midterm scores and could be significantly different from your final grade.

Late homework will not be accepted.

DSS advisory:

If you have a physical, psychological, medical, or learning disability that may affect your course work, please contact Disability Support Services (DSS) office: ECC (Educational Communications Center) Building, room 128, telephone (631) 632-6748/TDD. DSS will determine with you what accommodations are necessary and appropriate. Arrangements should be made early in the semester (before the first exam) so that your needs can be accommodated. All information and documentation of disability is confidential. Students requiring emergency evacuation are encouraged to discuss their needs with their professors and DSS. For procedures and information, go to the following web site http://www.ehs.sunysb.edu and search Fire safety and Evacuation and Disabilities.