18.786 (Topics in Algebraic Number Theory) Syllabus

The course in a nutshell

This course is a first course in algebraic number theory. Topics to be covered include number fields, class numbers, Dirichlet's units theorem, cyclotomic fields, local fields, valuations, decomposition and inertia groups, ramification, basic analytic methods, and basic class field theory. An additional theme running throughout the course will be the use of computer algebra to investigate number-theoretic questions; this theme will appear primarily in the problem sets.

A more precise breakdown of topics will manifest over the course of the semester on the lecture calendar.

Personnel

Lecturer: Kiran Kedlaya, 2-165, x3-2946, kedlaya[at]mit[dot]edu (more contact info)
Grader: TBA

Meeting times and places

Lecture: Tuesday/Thursday 11-12:30, room 2-102
Office hours: Wednesday 1-2, or by appointment, or drop-in

Textbook

Required: Algebraic Number Fields (second edition), by Gerald Janusz, published by the American Mathematical Society. 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 ($48 list price). It can be ordered online from the AMS Bookstore; it should also be available from the MIT Coop and/or Quantum Books.

In addition, I plan to post occasional course notes on topics not covered in adequate detail in Janusz; these will be linked from the lecture calendar.

Recommended (updated 7 Feb 06):

Murty and Esmonde, Problems in Algebraic Number Theory, is a good source of exercises, though I'm told it has a lot of errata.
I may dip into Silverman, The Arithmetic of Elliptic Curves at times; an even more basic introduction is Silverman and Tate, Rational Points on Elliptic Curves.
Neukirch, Algebraic Number Theory 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, An Invitation to Arithmetic Geometry.
For modular forms, try Diamond and Shurman, A First Course in Modular Forms.
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 and grading

Homework assignments will be given approximately weekly; I am currently planning to have them be due mostly on Thursdays. There will be one in-class midterm (probably on March 14) and a take-home final exam (probably to be given out May 16 and submitted May 23).

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 Google). 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.

Grading breakdown: Homework 50%, midterm 20%, final 30%.

Prerequisites

Elementary number theory (18.781).
Abstract algebra, including groups, rings and ideals, fields, and Galois theory (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 (18.100B). Real analysis in several variables (18.101) and complex analysis in one 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 (e.g., on registration day after 1 PM).