Number Theory – Mathematics 2530
Brown University – Fall, 2018
Professor Joseph Silverman

Text I'll mostly be using the following two books, but it's up to you whether you want to buy copies. Homework will be posted online. (I've listed the current Amazon prices. The Samuel book is very inexpensive.)
  • Algebraic Number Theory (2nd ed.), Serge Lang, Springer GTM, 1994. ISBN-13: 978-1461269229 ($57.11 hardcover, $69.95 paperback)
  • Algebraic Theory of Numbers: Pierre Samuel, Dover Books on Mathematics, 2008. (Translated from the French by Allan J. Silberger) ISBN-13: 978-0486466668. ($7.93)
But there are lots of good books that cover basic algebraic number theory in greater or lesser detail.
Office Mathematics Department, Kassar House, Room 202
Phone 863-1124
Web Site
Office Hours Monday 10:15–11:00 (none on Sept 10, Oct 8, Nov 12)
Thursday 9:00–10:00 (none on Nov 22)
(Or send me an email to make an appointment. I tend to be in on MTuTh and not on WF.)
Course Time TuTh 2:30–3:50pm (K hour)
Course Location Kassar 205
Prerequisites Math 2510–2520 or equivalent
Homework Homework assignments will be posted here. (I'll also try to announce them in class.)
Click here for web page containing the Math 2530 problems.
# Assigned Due Problems
1 Thur Sep 6 Thur Sep 13 1, 2, 3, 4, 5, 6, 7
2 Tues Sep 18 Thur Sep 27 8-I, 8-II, 9, 10, 11, 12, 13, 14
3 Tues Oct 2 Tues Oct 9 16, 17, 18, 20, 22, 23
4 Thur Oct 11 Thur Oct 18 24, 25, 26, 27, 28
5 Tues Oct 24 Tues Oct 30 29-I, 29-II, 30, 31
6 Thur Nov 1 Thur Nov 8 32, 33, 34, 35, 36
7 Tues Nov 13 Tues Nov 20 37, 38, 39, 40, 41
8 Tues Nov 27 Tues Dec 4 42, 43, 44
9 Tues Dec 4 Tues Dec 11 45, 46, 47, 48
Note on Using Computers in Math 253 I tend to use a computer program called PARI-GP to do number theory calculations. The good news about PARI is that it is free and very fast and powerful at doing number theoretic computations. The bad news is that it's not tremendouly user friendly, and also somewhat limited at doing calculations in algebraic number fields. You can download PARI by clicking here. An alternative to PARI is SAGE, which is also free. You can get SAGE at the SAGE web site. (You can also run PARI from within SAGE.)

Course Goals:

To learn the fundamentals of algebraic number theory, which forms the basis for all modern study of number theory and related topics.

Learning Activities and Time Allocation:

Learning activities include class attendance, frequent problem sets, and a take-home final exam. The time to complete these activities are (1) attending lectures, approximately 3 hours/week; (2) working on the problem sets and the final exam, approximately 9 hours/week.


Course grades will be determined by the quantity and quality of problem sets submitted (80% of grade) and by their grade on the takehome final exam (20% of grade).

Expectations of Students:

It is expected that students will attend all lectures and participate in class discussion in an appropriate manner. Assignments are due on the listed dates. All students are expected to abide by Brown's academic code, which may found here

Tentative Syllabus:

  1. Algebraic Integers, Dedekind Domains, Unique Factorization
  2. Completions
  3. Ramification, Discriminants
  4. Ideal Class Groups
  5. Unit Groups, Dirichlet's Theorem
  6. Galois Theory, Decomposition and Inertia Groups, Splitting of Primes
    Additional Topics (as time permits) chosen from:
  7. Cyclotomic Fields
  8. Adeles and Ideles
  9. Zeta Functions and L-series
  10. Tchebotarev Density Theorem

Academic Support: Brown University is committed to full inclusion of all students. Please inform me early in the term if you have a disability or other conditions that might require accommodations or modification of any of these course procedures. You may speak with me after class or during office hours. For more information, please contact Student and Employee Accessibility Services at 401-863-9588 or

Go to Professor Silverman's Home Page.