Math 8410 -- Number Theory II: TuTh 12:30-1:45, 326 Boyd
Instructor: Assistant Professor Pete L.
Clark, pete (at) math (at) uga (dot) edu
Course webpage:
http://math.uga.edu/~pete/MATH8410.html (i.e., right here)
Office Hours: During an hour long discussion section each week in
which all students are strongly encouraged to attend. Otherwise by appointment.
I am quite amenable to booking ``extra'' office hours. The ground rules
are: (i) please give me at least 24 hours notice. (ii) Please send me an email
the night before a morning appointment or the morning of a later appointment to
remind me that we are meeting. (iii) If I do not show for an appointment
(empirically, the chance that I will fail to show seems to be about 5-10%), feel
free to call me on my cell phone. Probably I'm not too far away. (iv) If we do
book an appointment, please do show up or call or email to let me know you're
not coming!
Discussion
Section: I would like to have an extra discussion section, one hour a week,
for discussion of problems and presentation of projects.
How Your
Grade is Computed: From homework, which may include a final project. Stay tuned.
Course Prerequisites: Math 8400 (Number Theory 1), or equivalent
knowledge. I will also draw on your general working knowledge of basic graduate level
mathematics, including field theory, topology and analysis. I do not expect a seamless
mastery of these subjects (and I hope you will extend the same courtesy to me!) but rather
that you be enthusiastic and proactive about improving your knowledge in these areas when
opportunities arise. Of course I will answer all questions to the best of my ability, including those on "background" material.
Course
Content: Here is a rough outline:
Part I: Absolute Values and Local Fields
Basic theory of absolute values, including Artin-Whaples Approximation
Completion of normed fields
Extension of absolute values
Structure theory of locally compact fields: unramified, tamely ramified and wildly
ramified extensions
Statements of local class field theory
Overview of Pontrjagin duality; self-duality of the additive group of a local field
Part II: Adeles and applications
Restricted direct product topologies
The adele ring
Adelic proofs of finiteness of class number and the Dirichlet unit theorem
Statements of global class field theory in the adelic language
Adelic points on algebraic varieties; weak and strong approximation
Course Notes (122 pages)
Chapter 1 (16 pages): Basic Theory of Absolute Values and Valuations (pdf)
Includes 22 problems.
Chapter 2 (29 pages): Completions and the Extension Problem
(pdf)
Includes 41 problems.
Chapter 3 (17 pages): The Fundamental In/Equality, Hensel and Krasner
(pdf)
Includes 25 problems.
Chapter 4 (10 pages): Structure Theory of Complete Discrete Valuation Fields
(pdf)