Math 204: Topics in Analytic Number Theory (winter 2013)

Course description

This year MATH 204 and 205 will be coordinated and cover the first 3 chapters of Cassels and Fröhlich. The sequence will provide a solid foundation in local and global fields. MATH 204 will cover roughly the first chapter of the book.
Cristian Popescu will teach MATH 205 in the Spring quarter and cover the other two chapters.

Instructor:	Alina Bucur (alina at math dot ucsd dot edu), APM 7131
Lectures:	MWF 10-11am, APM 7241
Office hours:	Tu 1-2pm and by appointment
Textbook:	Our main sources will be Algebraic Number Theory by Cassels and Fröhlich and the lecture notes. I will also provide additional references, should it become necessary.
Prerequisites:	I will only assume familiarity with Galois theory as acquired in an advanced level undergraduate course. In particular, this course should be accessible to first year graduate students.
Grading:	100% homework. Problem sets will be assigned weekly; please do them! It is effectively impossible to learn this subject passively. You are encouraged to work together, discuss the problems, etc; but the actual solution write-ups must be your own.

Announcements

No lecture Friday, January 18. Makeup lecture will be scheduled later.
There will be no office hours on Tu 1/22. Instead, I will hold office hours Th 1/24, 4-5pm. Since this is not the usual time slot, I realize this might not work with everybody's schedule. Please don't hesitate to email to either ask questions or to schedule an appointment at another time.
Typos in HW2 corrected. It does not mean it's typo-free, so feel free to signal additional issues.
Typos in HW4 corrected. It does not mean it's typo-free, so feel free to signal additional issues.
Office hours on Tuesday 2/5 have been rescheduled at 2:30-3:30pm.
Makeup lecture on Monday 2/18, 10-11am in APM 6402.
Office hours on Tuesday 2/19 have been rescheduled for 3-4pm.
HW6 corrected. It does not mean it's error-free, so please signal additional issues.
Office hours on Tuesday 2/26 have been rescheduled for 3-4pm.
No notes will be provided for the rest of the quarter. You will have to rely on the notes you take in class yourselves.
Office hours on Tuesday 3/12 are cancelled. Instead I will have office hours on Wednesday or Thursday this week (TBD).
Office hours this week will be Thursday 3/14, 11:30am-12:30pm.
The notes have been updated.

Lectures

The notes for the entire quarter can be found here.

Lecture 1 (M 1/7): generalized absolute values (definition, induced topology, equivalence)
Lecture 2 (W 1/9): any generalized absolute value is equivalent to a bona fide absolute value; non-archimedean and archimedean absolute values (definition, examples, a few properties), the p-adic absolute value
Lecture 3 (F 1/11): Ostrowski's theorem
Lecture 4 (M 1/14): completions
Lecture 5 (W 1/16): p-adic numbers and expansions in base p
Lecture 6 (W 1/23): p-adic digit expansion, p-adic integers, p-adic units, convergence, algebraic operations and square roots in Q_p
Lecture 7 (F 1/25): topology in the ring Z_p, Hensel's Lemma (part I)
Lecture 8 (M 1/28): Hensel's Lemma (part II)
Lecture 9 (W 1/30): applications of Hensel's Lemma, more about square roots in Q_p
Lecture 10 (F 2/1): the structure of the p-adic untis; definition of fractional ideal
Lecture 11 (M 2/4): fractional ideals, invertible submodules, invertibility is a local property
Lecture 12 (W 2/6): discrete valuations: definition, examples, connection to absolute values, properties; valuation ring, valuation ideal, residue field associated to a discrete valuation (definition and examples)
Lecture 13 (F 2/8): discrete valuation rings: definition, associated discrete valuation, all dvr's are valuation rings for some discrete valuation, structure of fractional ideals
Lecture 14 (M 2/11): characterization of dvr's, structure of the additive and multiplicative groups of the fraction field of a dvr
Lecture 15 (W 2/13): mth and pth roots in dvr's; Dedekind domains (def and examples)
Lecture 16 (F 2/15): equivalent characterizations of Dedekind domains, the ideal group of a Dedekind domain
Lecture 17 (M 2/18): the structure of the ideal group of a Dedekind domain, modules that span vector spaces
Lecture 18 (W 2/20): module index
Lecture 19 (F 2/22): the discriminant of a module
Lecture 20 (M 2/25): integrality and prime ideals
Lecture 21 (W 2/27): Dedekind domain in finite separable field extensions
Lecture 22 (F 3/1): local fields (def, examples, local compactness)
Lecture 23 (M 3/4): every local field is a finite extension of Q_p or F_q((t)), tensor product of fields
Lecture 24 (W 3/6): places of a field, extensions of generalized absolute values in the case of a complete field
Lecture 25 (F 3/8): places in finite separable extensions for an arbitrary field (not necessarily complete) and integral closures
Lecture 26 (M 3/11): ideal norms
Lecture 27 (W 3/13): different and discriminant of a field extension
Lecture 28 (F 3/15): ramification and inertia indices

Homework

Homework 1, due Wednesday, January 16 (pdf)
Homework 2, due Friday, January 25 (pdf)
Homework 3, due Wednesday, January 30 (pdf)
Homework 4, due Wednesday, February 6 (pdf)
Homework 5, due Wednesday, February 13 (pdf)
Homework 6, due Wednesday, February 20 (pdf)
Homework 7, due Wednesday, February 27 (pdf)
Homework 8, due Wednesday, March 6 (pdf)
Homework 9, due Friday, March 15 (pdf)