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 numbertheoretic
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,
2165, x32946, kedlaya[at]mit[dot]edu
(more contact info)
Grader: TBA
Meeting times and places
Lecture: Tuesday/Thursday 1112:30, room 2102
Office hours: Wednesday 12, or by appointment, or dropin
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 inclass midterm (probably on
March 14) and a takehome 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 takehome 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 graduatelevel course and will be
taught as such. That means I will expect a level of scholarly and mathematical
maturity appropriate to a firstyear 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).