18.786 | Spring 2006 | Graduate

Topics in Algebraic Number Theory


Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session



Janusz, Gerald J. Algebraic Number Fields. 2nd ed. Providence, RI: American Mathematical Society, 1996. ISBN: 0821804294.
This book has several advantages: It presents the material in the fashion that I think is most natural, it has reasonable examples and exercises, and it is not expensive.

In addition, I plan to provide occasional course notes on topics not covered in adequate detail in Janusz.

Murty, M. Ram, and Jody Esmonde. Problems in Algebraic Number Theory. 2nd ed. New York, NY: Springer, c2005. ISBN: 0387221824.
This is a good source of exercises.

I may dip into the following two books by Silverman:

Silverman, Joseph H. The Arithmetic of Elliptic Curves. New York, NY: Springer-Verlag, c1986. ISBN: 0387962034.

Silverman, Joseph H., and John Tate. Rational Points on Elliptic Curves . New York, NY: Springer-Verlag, c1992. ISBN: 0387978259. 

Neukirch, Jürgen. Algebraische Zahlentheorie. (Algebraic Number Theory). Translated from the German by Norbert Schappacher. Berlin, Germany; New York, NY: Springer, c1999. ISBN: 3540653996.
This is a text I have taught from before, but it is unfortunately very expensive. It also assumes more comfort with commutative algebra (and related ideas from algebraic geometry) than one might like.

For arithmetic geometry (like curves over finite fields), try: Lorenzini, Dino. An Invitation to Arithmetic Geometry. Providence, RI: American Mathematical Society, 1996. ISBN: 0821802674.

For modular forms, try: Diamond, Fred, and Jerry Shurman. A First Course in Modular Forms. New York, NY: Springer, c2005. ISBN: 038723229X.

There are lots of useful course notes available from James Milne’s Web site: Look for “Algebraic Number Theory,” and perhaps “Class Field Theory.”

Course Requirements

Homework assignments will be given approximately weekly. There is one in-class midterm and a take-home final exam. The midterm is detachable, meaning at least 2 of the four problems are due during the class period and the remainder are due at a later date.

As usual, you are encouraged to work on the homework in groups, but you must write up your own solutions, and I would like you to specify on your homework who was in your working group. On the take-home exam, you are to work on your own using only the specified resources (the book, your course notes, any book from the library, but not any human and not GoogleTM). In case of prima facie evidence of academic dishonesty, I reserve the right to ask you to defend your solutions, so don’t tempt me.


Homework 50%
Midterm Exam 20%
Final Exam 30%


Elementary number theory, e.g. Theory of Numbers (18.781). Abstract algebra, including groups, rings and ideals, fields, and Galois theory; e.g. Algebra I and II (18.701 and 18.702). A tiny bit of commutative algebra (18.705) may also help at times, mainly at the beginning. Real analysis in one variable (Analysis I, 18.100B). Real analysis in several variables (Analysis II, 18.101) and complex analysis in one variable (Functions of a Complex Variable, 18.112) may also help at times, mainly at the end.

All course numbers in the above list should be followed by “or equivalent”; I am the sole arbiter of what constitutes an acceptable equivalent. I’ll be particularly flexible about 18.781; if you studied number theory for an Olympiad, or in a high school summer camp, then you know what you need. I will be somewhat less flexible about 18.702; be prepared to convince me!

Special Notice for Undergraduates

Number theory is a popular topic, and so I expect there will be many undergraduates interested in this course; this means I need to provide a warning for such students. (There may also be some graduate students in other subjects without a full undergraduate math major; the same notice applies.) This course is listed as a graduate-level course and will be taught as such. That means I will expect a level of scholarly and mathematical maturity appropriate to a first-year graduate student in mathematics. In particular, material will go somewhat quickly, and you will be expected to pick up some of it on your own. Problem sets will be challenging; you will be expected to cope with this in appropriate ways, such as forming study groups. Basically, if you take this class, I’m going to treat you like a graduate student whether you are one or not.

All the scary stuff aside, any undergraduate with the relevant background is welcome to take the course; however, if you are using the “equivalent” option for any of the prerequisites, you need to have that cleared by me in writing (e.g., by email) or in person.

Course Info

As Taught In
Spring 2006
Learning Resource Types
Problem Sets