Math 748 - Algebraic Number Theory - Fall 2014

Instructor: Melanie Matchett Wood


The lecture schedule is posted below.

Office Hours

Course Webpage:

Course Description: An introductory graduate level course on algebraic number theory. Topics: a rigorous introduction to the arithmetic of number fields, unique factorization, algebraic integers, Dedekind domains and factorization of ideals, geometry of numbers, Dirichlet's Unit Theorem, ideal class groups, first case of Fermat's Last Theorem, local fields.

Prerequistes: a one-year course on Abstract Algebra at the graduate level, including various standard facts about groups, rings, fields, vector spaces, modules, and Galois Theory.

Text: Milne's Algebraic Number Theory Notes

Quick links

Homework assignments:

Homework assignments will be due each Tuesday at the start of class (paper copies must be handed in). Homework will not be accepted late (this is for your benefit so you keep up with the lectures). The final version will be posted by the preceeding Thursday and noon and say FINAL at the top of the pdf. (Earlier versions will be posted if I decide some problems early.) You may work together, but each student should do their own writeup of each problem. The problems from Milne have hints and/or solutions in the back of the text (though obviously for your own sake, you should do the problems without reference to these). Some of the homework will require sage computations (see below) . For the sage problems, please print some readable version of your sage terminal window, worksheet, or code and output, and indicate with written notes what the answer is.
  1. Homework 1 (due Sep 9)
  2. Homework 2 (due Sep 16)
  3. Homework 3 (due Sep 23)
  4. Homework 4 (due Sep 30)
  5. Homework 5 (due Oct 7)
  6. Homework 6 (due Oct 14)
  7. Homework 7 (due Oct 21)
  8. Homework 8 (due Oct 28)
  9. Homework 9 (due Nov 4)
  10. Homework 10 (due Nov 11)
  11. Homework 11 (due Nov 18)
  12. Homework 12 (due Dec 2--note week skipped)
  13. Homework 13 (due Dec 9)
  14. No more homework!


Sage is open source mathematics software that is, in particular, commonly used by number theorists. You can find a tour, tutorial, and further introductory material on sage on the sage website. As an introduction you could also look at the introduction and number fields sections of these lectures on using sage for number theory . There are lots of ways to use sage: in a terminal, in a browser worksheet, in sagemathcloud. You can pick whatever works best for you.

Lalit Jain has created a nice Sage/Linux quickstart reference which is available here. (Also perhaps see his documents on sage programming and using rings in sage.

Here is the Sage reference manual (in particular the section on "Algebraic Number Fields" will be very useful).

Other texts:

In this course, we will follow Milne's notes closely. There are many other great resources for learning algebraic number theory, and you might also use a complementary text if you find you need more explanation, or examples, or just another point of view. Some books I recommend include Marcus's ``Number Fields'' and Janusz's ``Algebraic Number Fields.'' Lang's ``Algebraic Number Theory'' and Neukirch's ``Algebraic Number Theory'' are also standard references. Some other online notes include

Lecture schedule: