Algebraic Number Theory: Spring 2008



Day/Time
         MW 1:10pm-2:25pm

Location
         307 Mathematics Building

Instructor
Christophe Breuil
Office Hours
Wednesday 4.00pm-6.00pm in Math 415
Contacting me
I only meet students during my office hours (unless the issue is very urgent) and I do not answer emails

Text
          Number Fields, by Daniel A. Marcus, Springer-Verlag

Teaching Assistant
Ming-Lun Hsieh (help room hours: Wednesday 4.00pm-7.00pm in Math 406)

Midterm exams
There will be (probably) two midterm exams. Make-up exams will not be given. Students will only be excused from the midterms because of a serious illness or another emergency of similar gravity, and a note from a doctor or a dean will be required
Final exam
May 12, 2008, 1.10 pm-4.00 pm, Math 307
All students must take the final at the time scheduled by the university
Homework
There are weekly homework assignments, mainly exercises from Marcus' book. Homework will be collected at the beginning of the Monday class. Late homework will not be accepted. Students are encouraged to discuss the homework with other students but should write their solutions individually
Grading
         Homework 20%
         Midterms 20% (each)
         Final exam 40%

Conflicts
 If you have a conflict with any of the exams (for example, due to a religious holiday), please contact instructor or T.A. as soon as possible 
Help Room
 If you would like help with the material, in addition to the office hours you can take advantage of the Math Help Room (Math 406). No appointment is necessary.

Tentative schedule of lectures

This schedule may be modified during the course. Partial solutions for the homework exercises will be available at:          
http://www.math.columbia.edu/~hsieh/teaching/alg-hw.html


Date

Reading

Homework

  Jan. 23

  Introduction

  Jan. 28, 30

  Commutative rings and ideals, Galois theory for subfields of C

  Ex.7,8,14 p.7-8
  Ex. 30,31,32 p.11 (all due Feb. 4)

  Feb. 4, 6

   Number fields and their ring of integers I

  Ex.1,2,4,5,6,11,25 p.39-45
  Ex.13,14,15,16,17,19,22 p.41-43 (all due Feb. 11)

  Feb. 11, 13

  Number fields and their ring of integers II

  Ex.23,26,27,28,29,42 p.43-53
  Ex.31,34,35,40,44 p.47-53 (all due Feb. 18)

  Feb. 18, 20

  Prime decomposition I

  Ex.7,8,12 p.83-84
  Ex.9,13,31 p.83-84 and 91-92 (all due Feb. 25)

  Feb. 25, 27

   Prime decomposition II
   Ex.20,24,28 except (c),(d) p.87-90
  Ex.14,19,32 p.84-92 (all due Mar. 3)

  Mar. 3

                                   Midterm 1                     Solution         

 

  Mar. 5
   Prime decomposition III
   Ex.17,26,29  p.86-90 (due Mar. 10)
  Mar. 10, 12
   Decomposition groups I
   Ex.7,8,14 p.114-119
  Ex.5,6,18,19,20,21,22,23,24 p.115-122 (all due March 24)

  Mar. 24, 26

  Decomposition groups II

  Ex.15,16
  Ex.10,11,12,13 (all due March 31)

  Mar. 31, Apr. 2

  Ideal class group and unit group I

  Ex.2,3,7,8
  Ex.13,14,25,26,28 (all due Apr. 7)

  Apr. 7, 9

  Ideal class group and unit group II

  Ex.31,32
  Ex.33,46 (all due Apr. 14)

  Apr. 14

                                      Midterm 2                     Solution


  Apr. 16
    Class number formula (without proofs)

  Apr. 21, 23

   Class number formula / Some class field theory (without proofs)

 

  Apr. 28, 30

   Some class field theory (without proofs)

 

  May 5

   Review 

 

  May 12
    Final exam