Math 204: Topics in Analytic Number Theory (winter 2013)
- 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.
|| Alina Bucur (alina at math dot ucsd dot edu), APM 7131
|| MWF 10-11am, APM 7241
| Office hours:
| Tu 1-2pm
and by appointment
| 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.
| 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.
| 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.
- 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.
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 Qp
- Lecture 7 (F 1/25): topology in the ring Zp, 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 Qp
- 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 Qp or Fq((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 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)