MATHEMATICS W4043, Fall 2015

Algebraic Number Theory



Time:  2:40-3:55, Tuesday and Thursday

Instructor:  Michael Harris

Office Hours:  Tuesday and Thursday, 10-11 and by appointment, room number 521


Teaching Assistant: TBA





This is an introduction to the algebraic theory of numbers.   The fundamental

techniques of the subject will be accompanied by the study of examples

of families of Diophantine equations that motivated the development of the subject.


Each of the topics listed below will occupy roughly two weeks of course time:


1.  Algebraic integers, factorization, Dedekind rings, local rings

            (Gauss's first proof of quadratic reciprocity)

2.  Units and class groups (Pell's equation, classification of binary quadratic forms)

3.  Cyclotomic fields (Fermat's last theorem for regular primes, first case;

            Gauss's fourth proof of quadratic reciprocity)

4.  Congruences and p-adic numbers (the Chevalley-Warning theorem)

5.  Zeta and L-functions (Dirichlet's theorem on primes in an arithmetic progression)

6.  Other topics (depending on class interest:  Dirichlet's unit theorem, the prime number theorem,

            cubic equations…)


Prerequisites:  Basic algebra through Galois theory.  Some elements of complex

            analysis may be admitted in section 5.



Textbook:  Marc Hindry, Arithmetics (Springer, 2011 edition)

Other useful references include


            Dan Flath, Introduction to Number Theory

            Pierre Samuel, Algebraic Theory of Numbers

            Jean-Pierre Serre, A Course in Arithmetic


The grade will be based on homework (20%), the midterm (30%), and the take-home final (50%).   


Midterm:  October 29

Final:  FINAL EXAM (online December 14, due December 21)



            Homework assignments

1st week  (due September 17)      

2nd week (due September 24)      

3rd week (due October 1)            

4th week (due October 8)

5th week (due October 15)

6th week (due October 22)

             (Midterm:  no homework)

7th week (due November 5)

8th week (due November 12)

9th week (due November 19 )

10th week (due December 3)

11th week (due December 10)



The final from the fall of 2013 is available here.


Course notes (will be frequently updated)





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