Algebraic Number Theory (Math 8630), Spring
Instructor: Andrew Obus
email: obus [at]
office: Kerchof Hall 208
TuTh 2:00-3:15, Monroe
Hall 114. Please ask questions if anything in lecture is unclear.
Lectures will run the entire 75 minutes. Please show up on time!
There will be a (roughly) biweekly one-hour problem session
will present homework solutions on the board, and where we can discuss
the solutions. The meeting time is Wednesday 5-6. Room is TBA.
Algebraic Number Theory, by Jürgen Neukirch. There
should also be a copy on reserve in the
library. This book is a "modern classic" in the field, in that it
takes great care to keep an algebro-geometric perspective throughout
(while still being accessible to those with no background in algebraic
geometry). The book contains much more material than we will be
able to study, and will be a good reference for future work. It
also contains lots of exercises. There is a bit of nonstandard
terminology --- I will point this out as we encounter it.
Other references you might enjoy for more-or-less standard material
If you want references for more advanced topics (Iwasawa theory,
diophantine geometry, arithmetic dynamics, cohomology of number fields,
modular forms, etc.), please
talk to me!
- Local Fields (Corps Locaux),
by Jean-Pierre Serre. A more streamlined exposition of local class
field theory, including the basic local material we will cover.
Has a few topics not covered in Neukirch.
- Number Fields,
by Daniel Marcus. This book is written at a somewhat lower level
than Neukirch, and deals only with the global picture. However,
it is nicely written and has a huge
number of computational exercises. Good for practice!
- Algebraic Number Fields,
by Gerald J. Janusz. I'm not so familiar with this book, but
people seem to like it. Perhaps "Neukirch-lite" is the best
description. Does topics in a somewhat different order.
- Algebraic Number Theory,
by Serge Lang. Covers a huge amount in a relatively short
book. Includes analytic theory as well.
- Algebraic Number Theory, by J. W. S. Cassels
and A. Fröhlich. Covers similar material to Lang. Comprised
of separate chapters by many of the giants in the field. Contains
a reproduction of Tate's thesis on functional equations of L-functions.
- Basic Number Theory, by
André Weil. Similar material to Cassels
and Fröhlich, but using much more topology and less Galois
cohomology. Don't be fooled by the title --- this is a difficult
Number Theory is the study of algebraic number fields (fields that are
finite over Q). Much of it arose from efforts to prove Fermat's
Last Theorem. Even though this is a question about Q itself, it
turns out that answering it requires looking at "higher" algebraic
number fields. Many other problems in number theory (e.g.,
quadratic reciprocity, classifying the primes that are sums of two
squares), are made easier from this perspective as well. The
structure of algebraic number fields is also fascinating in its own
right, and offers many open questions. For instance, is
Z[\sqrt(d)] a unique factorization domain for infinitely many
We will cover most of the material in chapters I and II of
Neukirch. This includes all of the standard results (finiteness
of the ideal class group, structure of the unit group, Hensel's lemma,
ramification theory). We will not do this in order, and will
spend some time on the local
(p-adic) theory before jumping into the global theory (algebraic number
fields). I find that thinking of number fields in terms of their
localizations lends itself better to retention. If time permits,
we will cover a few more topics (in particular, adeles and idèles).
The prerequisite for this course is Algebra III. Most of the
material from that course is not so essential, but you should know what
a tensor product is. If you have only had Algebra I and II and
want to take the class, this should be OK. The main material from
Algebra I and II on rings, modules, and fields is indispensable; I do
not recommend trying to take the course concurrently with Algebra
II. Occasionally, I will use commutative algebra concepts such as
integrality, localization, completion, and valuations. These will
be recalled as needed.
Tu 11-12, W 10-11. Kerchof Hall 208 (my office).
If the times do not work for you, please make an appointment with me.
You should be prepared to discuss/present
homework solutions at the problem sessions, which will meet roughly
every other week. You are encouraged to
work together on the homework, although I have found that this is more
effective after having first spent some time thinking about the
problems alone. If
you are officially enrolled in the class and have not yet passed all
your qualifying exams (written and oral), then you will be responsible
for handing in written homework. I recommend TeXing
solutions (I still reference my grad school homework solutions
every once in a while). If you want me to look over an answer
(even if you are not obligated to hand the homework in), I am always
There will be 6 homework assignments over the course of
the semester. Some of the problems will be difficult! The
homework assignments will be posted on Collab.
There are no exams in this class.
All students with special needs requiring accommodations should
present the appropriate paperwork from the Student Disability Access
Center (SDAC). It is the student's responsibility to present this
paperwork in a timely fashion and follow up with the instructor about
the accommodations being offered. Accommodations for test-taking (e.g.,
extended time) should be arranged at least 5 business days before an
you have (anonymous) comments for me about teaching style or anything
related to the course, you can make them on the Collab page for the