This course will give an introduction to the theory of number fields, which are fundamental objects in algebraic number theory. We will cover the following topics:
- review of field extensions, algebraic numbers
- rings of integers, discriminants, integral bases
- examples: cyclotomic fields
- non-unique factorisation of algebraic integers, unique factorisation into prime ideals
- fractional ideals, class groups
- lattices and Minkowski's lemma, finiteness of the class group
- computations of the class number
- group of units of a number field
- Dedekind zeta functions, class number formula
Gaolis theory
| Dates | Notes |
|---|---|
| September 20 | Lecture 1 |
| September 23 | Lecture 2 |
| September 27 | Lecture 3 |
| September 30 | Lecture 4 |
| October 4 | Lecture 5 |
| October 7 | Lecture 6 |
| October 11 | Lecture 7 |
| October 14 | Lecture 8 |
| October 18 | Lecture 9 |
| October 21 | Lecture 10 |
| October 25 | Lecture 11 |
| October 28 | Lecture 12 |
| November 1 | Lecture 13 |
| November 4 | Lecture 14 |
| November 11 | Lecture 15 |
| November 15 | Lecture 16 |
| November 18 | Lecture 17 |
| November 22 | Lecture 18 |
| November 25 | Lecture 19 |
| November 29 | Lecture 20 |
| December 2 | Lecture 21 |
| December 6 | Lecture 22 |
| December 9 | Lecture 23 |
| December 13 | Lecture 24 |
| December 17 | Lecture 25 |
| Dates | Exercise sheets | Solutions |
|---|---|---|
| September 30 | Problems 1 | Solutions 1 |
| October 14 | Problems 2 | Solutions 2 |
| October 21 | Problems 3 | Solutions 3 |
| October 28 | Problems 4 | Solutions 4 |
| November 4 | Problems 5 | Solutions 5 |
| November 11 | Problems 6 | Solutions 6 |
| November 25 | Problems 7 | Solutions 7 |
| December 2 | Problems 8 | Solutions 8 |
| December 9 | Problems 9 | Solutions 9 |
| December 16 | Problems 10 | Solutions 10 |