Advanced Topics In Algebra
Algebraic Number Theory
Fall 2004
Mathematics W4043 section 001
Tues Thurs 2:40-3:55pm
520 Mathematics Building
http://www.math.columbia.edu/~glass/fall04/
Darren Glass
Office: 516 Mathematics Building
Office Phone: (212) 854-5135
glass@math.columbia.edu
The best way to get in touch with me is by email. I am an email addict.
Announcements
Check this part of the website regularly for announcements.
- 12/14/04 - here is the cheat sheet, in preliminary
form. what else do you want to see?
- 12/9/04 - I have posted below some practice questions for the final. I will
hold a review session on Wednesday 12/15 from 11:30-1:30.
- 12/6/04 - There appears to be a rumor going around that in addition to
proposing problems for the final I also want you to hand in solutions to the
problems you propose. This is not the case. Just hand in the problems.
- 11/29/04 - Here
is an article detailing the use of class groups in cryptography, as I
alluded to in class today.
- 11/23/04 - The last day to turn in 'extra credit' problems is also on December
7th, the day the final homework is due. Yogishwar and I need time to grade
everything and we would like to get it back to you so you can use it all to study
for the final exam.
- 11/23/04 - My normal Wednesday afternoon office hour is cancelled tomorrow due
to the Thanksgiving holiday. Enjoy your turkey/tofurkey/stuffing/pumpkin pie.
- 11/10/04 - The article "What is a Reciprocity Law" which I referred to
in class tomorrow can be found here
(though you'll need to be on a Columbia computer to read it). A similarly
recommended article, "What is class field theory", can be found here.
- 11/5/04 - Several of you have asked me for specific suggestions of
problems to do for "extra credit". Since you sort of used Ch. 3 #16 in
order to do #17, I would suggest trying the rest of #16. Also, the
sequence of problems about the different -- Chapter 3, #33-39 -- would be
good to look at.
- 11/1/04 - Due to the election day (vote!) holiday there will not be
class tomorrow (vote!). Instead of my normal office hour (vote!), I will
be in my office during what would be classtime (vote!), so roughly from
3-4. Drop by then. If you've voted. I will have my normal office hour
on Wednesday, by which time it will be too late to vote.
- 10/13/04 - The material I discussed yesterday on fractional ideals is
NOT in the Marcus textbook. If you want a cliffs notes version recapping
the definitions and such, check out Mathworld.
If you want more depth I would check out Hungerford's Algebra book
or the Alaca and Williams book.
- 10/12/04 - I have written up some solutions to some of the homework
problems. See below. From here on out homeworks will be due on
Thursdays, and you should really really not wait until the last minute to
start them as I will absolutely not accept late homeworks any more.
- 9/27/04 - There's a typo in Problem 2 of HW Set 2. It should be a
comma, not a period. It has now been corrected.
- 9/24/04 - My Wednesday office hour this coming week will be cancelled.
Instead, I will be in my office on Monday (the 27th) from 2-3 if you want
to chat.
Course Information
Content
Number Fields are finite extensions of the field of rational numbers.
Such fields can be useful even in solving problems which appear to involve
rational numbers, and they are also the traditional introduction into the
subject of algebraic number theory. In this course we will study
algebraic numbers and integers, unique factorization of ideals,
ramification, and prime decomposition in number rings. If time allows we
will also discuss p-adic numbers and the Dedekind Zeta function. The only
prerequisite is the modern algebra sequence.
Warning: I will be assuming you have had Modern Algebra, and in
particular the Galois Theory that comes at the end of Modern Algebra.
A list of what I have done each day can be found at this website
Textbook
The official required textbook for the course is Daniel Marcus's Number
Fields. Depending on what we have time to do at the end of the
semester, I will likely cover some material from Fernando Gouvea's P-adic
Numbers book as well. If you want another reference, I highly
recommend the new book Introudctory
Algebraic Number Theory by Alaca and Williams.
Office Hours and Other Help
Number Theory is not an easy subject, and it is almost certain that you
will need help during the semester. Please do not hesitate to take
advantage of the various forms of help that we provide. The Columbia Help
Room is Mathematics 406. You may go there anytime it is open to have
your questions answered. The TA for this course will be Yogishwar
Maharaj, and he is in the help room from 1-4 on Wednesdays.
Other office hours: I will be in my office to chat on Tuesdays from
1-2:30 and on Wednesdays from 2:30-3:30. You can also email me to make an
appointment for other times.
Grading
60% - Homework
25% - Final Exam
15% - Attendance and Participation
Final Exam
The final exam will be held on Thursday, December 16th, from 1:10pm-4pm. It will be
in our normal location, and will be closed book closed note closed everything. I do
reserve the right to put some hints or formulae or the like on the exam, to serve as
a "cheat-sheet" for everyone. If there are things you would particularly like to
see on said "cheat sheet", then please email me.
Homework
Homework will be assigned roughly once a week. You are encouraged to
discuss homework with each other, but you must turn in your own work. If
you aren't sure exactly where the boundary lies, please ask me rather than
make assumptions. Late homework will be penalized unless prior permission
has been obtained from the professor. All homework will be graded not
only on correctness, but also on how well you communicate and explain your
answers. Mathematics is a process, not a final answer, and your work
should reflect that.
One of the things I like about Marcus's book is it contains a large number
of excellent problems. In an ideal (to me) world, I would have the
luxury of assigning you every problem in the book. I will not do this,
but I do suggest you at least read and think for a few moments about the
problems I do not assign. If you write up your thoughts/solutions to
these probelms and give them to me, it will act favorably to your grade.
- Homework 1 (due Thursday, 9/16) - DVI PDF
- Homework 2 (due Tuesday, 9/27) - DVI PDF
- Homework 3 (due Tuesday, 10/12) - No DVI PDF
- Homework 4 (due Thursday, 10/21) - DVI PDF
- Homework 5 (due Thursday, 11/4) - DVI PDF
- Homework 6 (due Thursday, 11/18) - DVI PDF
- Homework 7 (due Tuesday, 12/7) - DVI PDF
I have written up sketches to solutions to some of the problems on the
homeworks (mostly the ones I wrote). They are a work in progress and can
be found here in DVI form and here in PDF form. If there are further problems
you wish to see the solutions to, let me know. Like any good band, I may
or may not take requests depending on my mood. DISCLAIMER: I make no
promise that these solutions would get very many points from Yogishwar,
and in fact would be ashamed if they did. They are sketches that need to
be filled out, but you should be able to take it from here.
Other Resources