MATH 699 : Algebraic Number Theory
Lecturer: Rob Harron
Spring 2015
Lectures: F 2:30pm–3:30pm or 4:00pm
Keller 402
Office Hours: Mon. 2:30pm–3:30pm & Wed. 2:30pm–3:30pm in my office (Keller 407)
Lecturer: Rob Harron
Spring 2015
Lectures: F 2:30pm–3:30pm or 4:00pm
Keller 402
Office Hours: Mon. 2:30pm–3:30pm & Wed. 2:30pm–3:30pm in my office (Keller 407)
Announcements:
- I've never found that there was one algebraic number theory book that really satisfied me. I've been taking material from various sources. For the most part, so far, I've been using two main sources: the excellent book Introductory algebraic number theory by Alaca and Williams, which has a pleasing amount of background material, is at a good level, and has an amazing amount of examples (I highly recommend this, rather obscure, book!), and (Ram) Murty and Esmonde's Problems in algebraic number theory, which provides a nice presentation of the material, in addition to its wealth of interesting and à propos exercises. At times, these two books don't quite go far enough. In those situations, I tend to go for Neukirch's Algebraic number theory as that is where I first learned this material. Other sources I've taken material from are Fröhlich and Taylor's Algebraic number theory, Milne's notes Algebraic number theory and Class field theory, Narkiewicz' Elementary and analytic theory of algebraic numbers, and Janusz' Algebraic number fields.
Course material:
| Lecture 1 | Introduction, motivation, Gaussian integers (Neukirch §I.1) |
Problem set 1 |
| Lecture 2 | Integral extensions, rings of integers, example: quadratic fields (Neukirch §I.2, Alaca–Williams Ch. 4, Atiyah–Macdonald Ch. 5) |
Problem set 3 |
| Lecture 3 | Discriminants, finding OK, example: pure cubic fields (Neukirch §I.2, Alaca–Williams §7.1, Murty–Esmonde §4.3) |
Problem set 4 |
| Lecture 4 | Discriminant and ring of integers of cyclotomic fields, Dedekind domains (Murty–Esmonde §4.5, Fröhlich–Taylor §II.1, Atiyah–Macdonald Ch. 9) |
Problem set 2 |
| Lecture 5 | Dedekind domains (cont'd), factorization in extensions (Neukirch §I.8) |
Problem set 5 |
| Lecture 6 | Discriminants and differents of extensions, Dedekind's theorem on ramification (Neukirch §III.2, Murty–Esmonde §5.4) |
Problem set 6 |
| Lecture 7 | Factorization in Galois extensions (Neukirch §I.9) |
Problem set 6 |
| Lecture 8 | Factorization in cyclotomic fields, a proof of quadratic reciprocity (Neukirch §I.10) |
Problem set 7 |
| Lecture 9 | Geometry of numbers, application to finiteness theorems (Neukirch §I.5, III.2, Alaca–Williams Ch. 12, Milne's ANT notes Ch. 4) |
Problem set 8 |
| Lecture 10 | More finiteness theorems, finiteness of the class number (Neukirch §I.6, III.2, Alaca–Williams Ch. 12, Milne's ANT notes Ch. 4) |
Examples of computing class groups |
| Lecture 11 | Examples of computing class numbers (Alaca–Williams §12.6) |
Problem set 9 |
| Lecture 12 | Dirichlet's unit theorem, the regulator, and units in quadratic fields (Alaca–Williams Ch. 11, Neukirch §I.7, Milne's ANT notes Ch. 5) |
Problem set 10 |