Algebraic Number Theory - DIAMANT / Mastermath 2012

- The notes are continually updated, but the changes are very mild. If you want a printed version of the notes, then you can print at any time. However, you may want to check whether the problem numbers are unchanged, for the homework. Please also write the date of the version of the notes that you used on your homework.
- Regarding registration: please register
**both**via the online form and by signing and correcting the paper list at the lectures.

See the Mastermath page.

Lectures: |
Peter Stevenhagen (Universiteit Leiden) Marco Streng (Vrije Universiteit) | |

Problem class: |
Athanasios Angelakis Julio Brau Pınar Kılıçer email address for the assistants: ant.mastermath.2012@gmail.com | |

Location: |
Tuesday 11 September: MF-A301 (different building: "MF", see below) Tuesday 18 September: MF-A311 (different building: "MF", see below) Tuesday 25 September: WN-M143 (back in the original building: "WN", but not the original room (P655), and not the room that was on the web page before (M129). The "M" part of the building is somewhere in the middle of the main (very long) hallway.) Tuesday 2 October: WN-S655 (the original room) Tuesday 9 October: WN-S655 (again) Tuesday 16 October: WN-M143 (same as 25 September) Tuesday 23 October: no class Tuesday 27 November: WN-P447 (computer room) Other lectures, unless specified otherwise: WN-S655 (where the first lecture was). " MF" means "Medische Faculteit" and the rooms A301 and A311 are on the third floor, right above the main entrance.
The arrow on this map
points to the main entrance." WN" means "Wis- en Natuurkundegebouw" (map), Vrije Universiteit, De Boelelaan 1081a, Amsterdam | |

Time: | Tuesdays, 10:15 -- 13:00,
consisting of 2x 45 minutes of lectures and 1x 45 minutes of problem class Tuesday 4th September 2012 -- Tuesday 11 December 2012 exception of 23 October: catch-up week |

We follow the lecture notes Number Rings by Peter Stevenhagen. They will be updated regularly, but the changes are small, so if you prefer a paper version, you can print it at any time.

If you feel you have gaps in your knowledge, you can look at the notes for Algebra 1,2,3 and/or consult an algebra textbook and/or ask us.

The final grade is exclusively based on the weekly homework assignments. The problems for this homework are announced in the timetable below, and you have some freedom in choosing the problems.

Solutions have to be handed at the start of the lecture where they are listed. This means they must be on a pile in the front of the lecture room when the lecture starts, or received at ant.mastermath.2012@gmail.com before the start of the lecture. Please write the date of the version of the notes that you used on your homework.

Working together is generally a good idea, but the final writing-down has to be done individually. Homework that is copied almost word-for-word will not be accepted (this is true for both the original and the copy).

It is generally a good idea to look at many exercises, not just the ones you will hand in, and see what they mean. For example, what is their relation with the class and with the surrounding exercises, and why did we set this problem? Try to see how one would go about solving them. And then work the requested number of problems out in detail and hand them in.

Easy choices, or choices with a lack of variation may result in lower marks. Homework that is late will result in lower marks (except possibly for reasons of illness).

Date | Lecturer | Subject | Exercises due | |
---|---|---|---|---|

1 | 4 September | both | Practical information, introduction, examples (Chapter 1) | (none) |

2 | 11 September | PS | Ideal arithmetic, fractional ideals, invertible ideals (Chapter 2, up to, but not including 2.6) | 4 exercises from 1.9 -- 1.36, of which at least one from 1.9 -- 1.12. |

3 | 18 September | both |
The Picard group, localization, orders (Chapter 2: 2.6 -- 2.13) |
4
exercises from 2.8 -- 2.19. You may choose to replace one (or even two if you make a very nice selection) problems by problems from the previous week (but not those you already handed in, and not those treated in class). For problems 2.14 -- 2.19, read Definition 2.6 and the text around it. |

4 | 25 September | both |
More on localization, local number rings, discrete valuation rings, Dedekind domains,
unique prime ideal factorization, the Kummer-Dedekind theorem,
examples (Chapters 2 and 3: 2.13 -- 3.2) |
4 problems from 15, 17, 18, 20 -- 34 (Chapter 2), of which at least one from 29 -- 34, and with the exception of those already handed in. |

5 | 2 October | MS |
Subject: examples, regularity when enlarging a number ring,
integrally closed (Chapter 3: up to and including 3.14) | 4 problems from 2.37 -- 2.59 and 3.8 -- 3.21, of which at least one from each of the two chapters. |

6 | 9 October | PS | Integral closure, normalization, cyclotomic rings (remainder of Chapter 3) |
4 exercises from 3.8 -- end of Chapter 3, but not those you have already handed in or those done in class |

7 | 16 October | MS | Norm, trace, characteristic polynomial, discriminant, the ring of integers is an order (Chapter 4: up to and including 4.8 and its proof) | 4 problems from 3.22 -- end of Chapter 3 (excluding 23 and 24, which were treated in the problem class, and excluding the problems you have already handed in) |

23 October | no class | |||

8 | 30 October | PS | Resultants, computing norms and discriminants, examples, ramification (remainder of Chapter 4) | 4 problems from 4.3 and 4.7 -- 4.15 |

9 | 6 November | MS | Minkowski's theorem, finiteness of the class group (Chapter 5) | 4 problems from 4.15 -- 4.32 |

10 | 13 November | both | Dirichlet Unit Theorem (remainder of Chapter 5), computing unit and class groups (selection from Chapter 7), the Dedekind zeta function and application to class numbers and regulators (Chapters 6) |
4 problems from 5.4 -- 5.34 reminder for the students and the assistants: as stated above, "easy choices, or choices with a lack of variation may result in lower marks" for example due to stricter grading of such choices |

11 | 20 November | MS | computing unit and class groups (Chapter 7) | additional problems |

12 | 27 November | MS |
Sage,
different location: computer room P447 Follow the tutorial you can find here https://sage.math.leidenuniv.nl/home/pub/13/edit_published_page | personal polynomial assignment, and to help you with that, you can have a look at these instructions for computing Euler products |

13 | 4 December | PS | The number field sieve and Galois theory. | Sage homework (here is a text file of the unit in problem 5 for convenience) |

14 | 11 December | MS | Some things that follow basic algebraic number theory. |
Final homework: 4 well-chosen problems from 8.5 -- 8.15. You can read and use whatever you need from chapter 8. Some of the problems are not in the notes yet, but can be found here |