Reading Course Agenda (Fall 2008)

[Chris Swierczewski]

There will be a reading course this Fall Quarter on some of the topics
in last year's Algebraic Number Theory series. The purpose of this
course is to prepare those attending for next quarter's Computational
Algebraic Number Theory course! Here are some notes on they key
concepts and textbooks taken from the agenda on the course wiki:

http://wiki.wstein.org/ant07
http://wiki.wstein.org/ant07/sched (course outline)

* Key Concepts:
1) Algebraic Integers
2) Commutative Algebra
3) Dedekind Domains and Rings
4) Group of Fractional Ideals
5) Unique Factorization of Ideals
6) Generalized Chinese Remainder Theorem
7) Discriminants and Ramification
8) Norms of Ideals and Class Groups (esp. important)
9) Finiteness of Class Groups (esp. important)
10) Unit Group Theorem (structure theorem of group of units of a
number field)
11) Decomposition of Prime Ideals (factorization and structure
theorems)
12) Decomposition Groups and Inertia Groups

* Things to Especially Omit
1) Algorithms for Factoring
2) Algorithms for just about anything (this is the key purpose of next
quarter)
3) Anything past Thanksgiving on last year's agenda.

* Books
1) The course textbook (http://sage.math.washington.edu/books/ant/)
2) "Number Fields" - Marcus
3) "Number Theory: Vol. I and II" - Henri Cohen
4) "A Course in Computational Algebraic Number Theory" - Henri Cohen

And finally, just as a reminder, our homework assignment is to read
and do as much of chapter two of the textbook as possible!