## 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

• Restricted direct product topologies
• 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)

Chapter 5 (10 pages): Locally Compact Fields (pdf) (version of 3/31/10)

Chapter 6 (8 pages): Adeles and Ideles (pdf)

Chapter 7 (7 pages): Idelic Approach to Class Groups and Unit Groups (pdf)

Chapter 8 (12 pages): Ray Class Groups and Ray Class Fields: First Classically, then Adelically (pdf)

Bonus Chapters:

Chapter 9 (13 pages): Applications of Local Fields (pdf)

• Extra Problems: (you do not need to solve these, but I might want to refer to them later and incorporate them into a later draft of the notes).

Exercise 1.23 [taken from an in-class discussion]: Suppose R is a Dedekind domain with fraction field K, p_1,...p_r are distinct nonzero prime ideals of R, n_1,..,n_r are integers, and x_1,..,x_r lie in R. Show that there exists an element x in K such that:
(i) for all 1 <= i <= r, v_{p_i}(x - x_i) = n_i, and
(ii) v_q(x) >= 0 for all nonzero primes q distinct from the p_i's.

Exercise 1.24 [taken from Endler, Valuation Theory, (3.9)]: Let (K,v) be a non-Archimedean valued field of residue characteristic p > 0. [Remark: The word "residue" was omitted when this problem was first posted, which makes the problem almost trivial.] Suppose that x and y both lie in the valuation ring. Show that v(x-y) > 0 implies v(xp - yp) > v(x-y).

Suggestion #1: Try to adapt the proof of Lemma 3 from the Math 4400/6400 number theory handout A Word About Primitive Roots
Suggestion #2: Show that for any prime number p and elements x,y in any commutative ring R, there exists an element c in R such that
xp - yp = (y + (x-y))p - yp = (pc + (x-y)p-1)(x-y).

Exercise 1.25 [essentially suggested by Alex Rice in class; taken from Endler, (3.13)] Let v_1,..,v_n be pairwise inequivalent valuations on a field K. For any x_1,...,x_r in K and gamma_i in Gamma_i = v_i(K^{\times}), there exists an x in K such that v_i(x-x_i) = gamma_i for all 1 <= i <= n.

Exercise 1.26 [asked by Alex Rice]: Let K be a normed field. Is it possible for the completion of K to be algebraic over K of infinite degree?

Exercise 2.?? [asked by Adrian Brunyate]: Does there exist an Archimedean [i.e., not non-Archimedean!] normed abelian group with the property that every convergent series is unconditionally convergent? Absolutely convergent? (Suggestion: Look at the standard absolute value on the integers.)

Exercise 2.?? [asked by David Krumm]: We know that if K is a complete normed field, then every linear map between finite dimensional normed linear K-spaces is continuous. What about the converse: if K is a normed field such that every linear map between finite dimensional K-spaces is continuous, must K be complete? (Suggestion: consider the algebraic closure of Q_p.)

Homework Solutions: If you spent a lot of time on any one problem, found it especially interesting or challenging, and/or presented the solution in the problem session, please consider texing up the solution, then send me the tex file. I will (possibly after mild editing, with your permission) post it here. If you do this, please name your tex file as follows: Math8410Ex[chapter number].[exercise number][your name here].

Solutions to Exercises 1.3, 1.12, 1.14, 1.16, by John Doyle (pdf)

Solution to Exercise 1.24, by Jim Stankewicz (pdf)

Solution to Exercise 2.4 (via transfinite induction), by Pete L. Clark (pdf)

David Krumm shows that a valuation extends to any extension field (in particular, this solves Exercise 2.4) (pdf)

Solution to Exercises 3.5, 3.9.5, 3.11, 3.13.5, by Kate Thompson (pdf)

Solution to Exercise 5.3, by Adrian Brunyate and Alex Rice (pdf)

Solution to Exercise 3.17, by David Krumm (pdf)

Jim Stankewicz classifies separable degree p extensions in characteristic p and gives applications to the failure of Hermite's finiteness theorem for global fields of positive characteristic (pdf)

David Krumm shows that every overring of a Dedekind domain is a localization iff the ideal class group is torsion (pdf)

OUR DISTINGUISHED COMPETITION (lecture notes by other people):

History of valuation theory, part I, by Peter Roquette: (pdf)
The completion of an algebraically closed normed field is algebraically closed, by Brian Conrad: (pdf)
Tame ramification and composite extensions, by Brian Conrad: (pdf)
In/finiteness of the number of degree n extensoins of a local field, by Brian Conrad: (pdf)
Global extensions approximating local extensions, by Brian Conrad: (pdf)
Adeles and Ideles, by J.W.S. Cassels: (pdf)
Compactness of the norm one idele class group (Fujisaki's Lemma), by Brian Conrad: (pdf)
Fujisaki's Lemma, by Paul Garrett: (pdf)
Topology of S-integer rings, by Brian Conrad: (pdf)
The idelic approach to number theory, by Tom Weston: (pdf)
Brief summary of the statements of class field theory, by Bjorn Poonen: (pdf)