Math 105: Algebraic number theory
Fall 2014
Course Info:
- Lectures: Monday, Wednesday, Friday, block 12 (12:30 p.m. - 1:35 p.m.)
- x-period: Tuesday, 1:00 - 1:50 p.m.
- Dates: 15 September 2014 - 17 November 2014
- Room: 004 Kemeny
- Instructor: John Voight
- Office: Kemeny Hall, Room 341
- E-mail: jvoight@gmail.com
- Office hours: Wednesday and Thursday, 9:00 - 10:30 a.m., or just make an appointment!
- Course Web Page: http://www.math.dartmouth.edu/~m105f14/
- Prerequisites: One year of abstract algebra (groups, rings, fields) at the advanced undergraduate or graduate level.
- Required Texts: Juergen Neukirch, Algebraic number theory, Grundlehren Math. Wiss., vol. 322, 1999.
- Grading: For undergraduates and graduate students in years 1 and 2, grade will be based on weekly homework.
Syllabus:
[PDF] Syllabus
This course will be a graduate-level introduction to algebraic number theory, in which we will cover the fundamentals of the subject. Topics may include: rings of integers, Dedekind domains, factorization of prime ideals, Galois theory in number fields, geometry of numbers and Minkowski's theorem, finiteness of the class number, Dirichlet's unit theorem, selected topics from analytic number theory, quadratic and cyclotomic fields, localization and local rings, valuations (i.e. p-adic) and completions, an introduction to class field theory, application to Diophantine equations, and other topics as time permits.
Homework:
The homework assignments will be assigned on a weekly basis and will be posted below. Homework is required for undergraduates and graduate students in years 1 or 2; it is optional but strongly encouraged for anyone else. In general, it is due in one week, but late homework will be accepted.
Cooperation on homework is permitted (and encouraged), but if you work together, do not take any paper away with you--in other words, you can share your thoughts (say on a blackboard), but you have to walk away with only your understanding. In particular, you must write the solution up on your own.
Plagiarism, collusion, or other violations of the Academic Honor Principle, after consultation, will be referred to the The Committee on Standards.
[PDF] Homework Submission Guidlines
| 1 | 15 Sep | (M) | Introduction, I.1 (Gaussian Integers) | |
| 2 | 17 Sep | (W) | I.2 (Integrality) | |
| 3 | 19 Sep | (F) | I.2 | |
| 4 | 22 Sep | (M) | I.2 | |
| 5 | 24 Sep | (W) | I.3 (Ideals) | I.1.3, I.2.3, I.2.6 |
| 6 | 26 Sep | (F) | I.3 | |
| 7 | 29 Sep | (M) | I.3, I.4 (Lattices) | |
| 8 | 30 Sep | (T) | I.4 (Lattices) | |
| 9 | 1 Oct | (W) | I.5 (Minkowski Theory) | |
| 10 | 3 Oct | (F) | I.5, I.6 (The Class Number) | I.3.1, I.3.5, I.3.8, I.3.9 |
| 11 | 6 Oct | (M) | I.6 | |
| 12 | 7 Oct | (T) | I.6 | |
| 13 | 8 Oct | (W) | I.7 (Dirichlet's Unit Theorem) | |
| 14 | 10 Oct | (F) | I.7, I.8 (Extensions of Dedekind Domains) | I.4.1, I.5.3 (may use I.5.2 without proof), I.6.3 |
| 15 | 13 Oct | (M) | I.8 | |
| 16 | 15 Oct | (W) | I.8, I.9 (Hilbert's Ramification Theory) | |
| 17 | 17 Oct | (F) | I.9, I.10 (Cyclotomic Fields) | |
| 18 | 20 Oct | (M) | I.10 | I.7.4, I.7.5, I.8.2, I.8.8 |
| 19 | 21 Oct | (T) | I.10 | |
| 20 | 22 Oct | (W) | I.11 (Localization) | |
| - | 24 Oct | (F) | No class (JV at NYU) | |
| 21 | 27 Oct | (M) | I.11, II.1 (The p-adics) | |
| 22 | 29 Oct | (W) | II.2 (The p-adics) | I.9.1, I.9.2 (typo: q=N(pp)), I.10.2 (assume I.10.1), I.11.6, I.11.A |
| 23 | 31 Oct | (F) | II.2 | |
| 24 | 3 Nov | (M) | II.2, II.3 (Valuations) | |
| 25 | 4 Nov | (T) | II.3 | |
| 26 | 5 Nov | (W) | II.4 (Completions) | |
| 27 | 7 Nov | (F) | II.5 (Local Fields) | II.1.1, II.1.5, II.2.4, II.3.3 |
| - | 10 Nov | (M) | No class (JV in Montreal) | |
| - | 12 Nov | (W) | No class (JV in Montreal) | |
| 28 | 14 Nov | (F) | ||
| 29 | 17 Nov | (M) |