Sylvain BONNOT  

MAT 311 
Number Theory 
 

We will meet on MWF : 10:40 am to 11:35 am in Physics P112.

First day of class: Monday January 22, 2006.
Final exam : TBA.

Office hours:
every Wedn. from 2:00 pm to 5:00 pm in my office, 5D-148 in the Math Tower.
My office is in the I.M.S (Institute for Math. Sciences), located on floor 5 and a half.

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

Our textbook:
(added monday 01/22): I confirm that our textbook will be the following: 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:

  • The final exam is now graded, and I posted your letter grades on the Solar system, you should be able to view them very soon. You did a rather good job for this final, that was not an easy one! Since I know you want to do some more number theory in the summer, I give you the correction of the final. I wish you good luck for the rest of your exams, and have a great summer!

  • NEW! Please try the practice final, and if you are stuck read its correction.

  • The correction for HW9 is available.

  • Here is a Practice Final that you should try. Ask me questions about it if you need to! The final exam is next week, on Wed.9th, 8am to 10:30am, usual room. It covers everything from the beginning.

  • Here is a beginning: some questions . Of course, you will have more on monday, but you already have to finish a HW, I guess?

  • I promised you the end of the proof of the fact that a periodic continued fraction corresponds to a quadratic irrational number. Also, the correction for HW8 is on the HW page.

  • HW9 is available on the HW page.

  • HW8 has been updated: I changed a small error in problem 2, and added some hints for the last problem.

  • HW8 is available.

  • You might want to check the correction: scan1, scan2,scan3. I don't give any homework for the break: have a well-deserved Springbreak and see you in April!

  • Your midterm will be graded during the break, and will be returned on wed. april 11th. If you need your grade earlier, send me an email. The monday class (april 9th) will be given by our grader Caner Koca, about continued fractions. We will use them to describe the invertible elements in quadratic fields.

  • About Midterm II I corrected the practice exam this morning: in case you need it, here is the file of the correction. The second midterm exam is this friday, usual room, usual time. Also the correction for HW7 is on the HW page.

  • You should try this practice exam.You will have a correction of it on monday. Prepare your questions for me! And remember that the exam is on friday 30th!!

  • The correction of HW6 is on the HW page, and HW7 has been posted: please notice that it is a much shorter one, that it is due on monday, and also that you are encouraged to ask questions about it!(you are also encouraged to return it)

  • The correction of the midterm and HW5 are available... correction of the midterm Also HW6 is on the HW page.

  • Midterm I is graded You did a pretty good job, the average is around 63/100... The next Hw assignment will be given this Friday (03/09).

  • Correction for the practice exam If you tried it, you may want to read its correction ...

  • Next HW assignment will be given on monday On monday we will review for the exam, so prepare your questions for me! The correction for the practice exam will be here very soon...You can bring your HW5 on monday too.

    • Midterm I is next week on Wednesday March 7th, usual room, usual time Please read these informations about what you should know for next week.

    Brand new practice exam !! Please try this practice midterm 1, and remember that you will have a detailed correction available very soon...

    New HW5 is on HW page.

    Please read this proof of the Quadratic reciprocity.Ask me questions if you need!

    The two midterms have been scheduled, please see below.

    The correction for HW3 is available on the HW page.

    Here is a new assignment: HW4.

    The correction of HW2 is now available.

    Here is the third homework assignment: HW3.

    Here is the second homework assignment: HW2.

    The correction of HW1 is now available.

    Here is the first homework assignment: HW1. Don't hesitate to ask me questions if something is not clear for you!

    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:January 22,24,26
    		 Divisibility, prime numbers,repartition of primes, rational points on circle
    Week 2:Jan. 29,31,Feb. 02
    		 Congruences, Euclid algorithm, linear equations
    Week 3:Feb. 5,7,9
    		 Euler's phi function, summary about groups,rings,Chinese remainder theorem
    Week 4:February 12,14,16
    		 Structure of the multiplicative group, existence of square roots
    Week 5:Feb. 19,21,23
    		 Quadratic reciprocity theorem
    Week 6:Feb. 26,28,March 02
    		 The RSA cryptosystem, Rabin's system, basic attacks on RSA
    Week 7:March 5,7,9
    		 Review,exam: midterm 1
    Week 8:March 12,14,16
    		 Ideals, quotient of a ring by an ideal, quadratic extensions
    Week 9:March 19,21,23
    		 Prime ideals (continued), basic intro to topological spaces,Spec of a ring
    Week 10:March 26,28,30
    		 Review, midterm II
    Week 11:April 9,11,13
    		 Continued fractions and approximations of real numbers
    Week 12:April 16,18,20
    		 Intro to elliptic functions and cryptography

    Exams:

    Midterm 1  Wed. March 7th Usual room
    Midterm 2   Fr. March 30th Usual room  
    Final   Wed May 9,2007, 8:00 am to 10:30 am 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%

    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.