Math 676 Syllabus
Math 676
Math 676 is a first semester
graduate course in algebraic number theory.
Course details
Time: MWF, 11AMnoon
Place: 4096 East Hall
First meeting: Wednesday, September 3, 2008
About the instructor
Jeffrey Lagarias
Email: lagarias@umich.edu
Web: http://www.math.lsa.umich.edu/~lagarias
Office: 3086 East Hall
Office Phone: 7631186
Office Hours: TBA
Course description
The fundamental theorem of arithmetic states that any nonzero integer
is expressible as a product of primes in a manner that is unique up
to order of the factors and units (i.e. signs). Algebraic number theory
begins with trying to understand how this generalizes (or fails to do so)
in algebraic number fields. An algebraic number field is a finite extension
of Q, and an element of such a field is called an algebraic number. The course
covers the basic structure of such field and some analogs in positive
characteristic. In particular we will study algebraic integers (these
play a role analogous to that of Z inside of Q) and we will prove two
of the important finiteness theorems: finiteness of class grous (these
measure the failure of the fundamental theorem of arithmetic for rings
of algebraic integers) and finitegeneratedness of unit
groups (these generalize classical results centered on Pell's equation).
We will include as many examples and applications as possible and some
of these will be developed in the homework. The main topics to be
covered are:
(1) Valuations and local fields
(2) Dedekind domains, global fields, rings of integers
(3) Class groups
(4) Unit groups
(5) Lfunctions and class number formulas
If time permits we may discuss the adele ring and idele group
attached to a number field.
The course (or at least a solid command of the material in it, ignoring
characteristic p if you are analytically inclined) is absolutely
essential for anyone wishing to study number theory.
Textbooks
The basic textbook the course will use is:
J. W. S. Cassels,
Local Fields,
Cambridge University Press 1986.
Students should get a copy of this textbook. It is available
in paperback.
Homework Assignments:
Grading is based on homework.
Homeworks are due approximately biweekly.
Students
are expected to in solutions to
5 problems out of total;
doing more problems is optional.
Homework 1
(due Wednesday, Sept. 17 )
Homework 2 (due
Wednesday, Oct. 1 )
Homework 3 (due Friday Oct. 17 )
Homework 3 (11th problem) (due Friday, Oct. 17 )
Homework 4 (due Wednesday, Nov. 5 )
Homework 4 (11th problemmandatory) (due Wednesday, Nov. 5 )
Homework 5 (due Wednesday, Nov. 19 )
Homework 6 (due Wednesday, Dec. 3 )
Homework 7 (due Monday, Dec. 8 )
Syllabus and Homeworks
Homeworks are approximately biweekly.
This syllabus will be updated as the course progresses.
Course Schedule/Homework Dates
Other Books
Other sources:
J. Milne,
Algebraic Number Theory, version 3.00 notes.
It is downloadable at
http://www.jmilne.org/math/index.html">
(Please make only one copy for personal use.)
These notes assume knowledge of commutative algebra.
A good, inexpensive, book to get is:
P. Samuel,
Algebraic Theory of Numbers
(Hermann, Paris 1970) Dover reprint 2008.
Another good book:
D. Marcus,
Number Fields ,
Springer_Verlag, New York 1977
This book has a great selection of exercises.
Other good references:
S. Lang,
Algebraic Number Theory (Second Edition)
SpringerVerlag: New York 1994
G.Janusz,
Algebraic Number Fields, 2nd Edition (AMS 2005)
A very good modern source is:
J. Neukirch,
Algebraic Number Theory (SpringerVerlag 1999expensive).
For extensions to class field theory:
 Cassels and Frohlich, Algebraic Number Theory (excellent, but
out of print)
 Serre, Local Fields (for local class field theory)

Washington, Introduction to Cyclotomic Fields (for KroneckerWeber,
and some applications)
Prerequisites
Graduate coursework in algebra
is also recommended.
If you are unsure how your
background matches these prerequisites, see me.
Homework/Exams
TBA
Jeff Lagarias
(lagarias(at)umich.edu)