# Algebraic Number TheoryWS 2021/2022

Dietrich Burde

Lectures: Monday 13.15 - 14:45 and Wednesday 13:15 - 14:45

• Number theory is an area in mathematics dealing with properties of integers in the broadest sense. Algebraic number theory studies the arithmetic of algebraic number fields, i.e., the ring of integers in the number field, the ideals and units in the ring of integers, the extent to which unique factorization holds, the geometry of numbers and further topics. Much of the theory arose out of the study of Diophantine equations.

The chapters are as follows:

Chapter 1: Integral ring extensions, in particular global fields and their rings of interegs, norm, trace and discriminant.
Chapter 2: Ideals in Dedekind rings, fractional ideals, ideal class group, unique factorization.
Chapter 3: Finiteness of the class number, Minkowski-Theory, rings of integers as lattices, special case calss number \$1\$ fields.
Chapter 4: Dirichlet's Unit Theorem, the analytic class number formula.
Chapter 5: Decomposition and ramification, in general and for Galois extensions. Ramification and discriminant.
Chapter 6: Cyclotomic fields and their rings of integers, units, and the Fermat equation.
Chapter 7: Absolute values and local fields, completions, the adelic viewpoint.
Chapter 8: The Theorem of Kronecker-Weber.

• Here are the dates for this lecture in u:find of the University of Vienna.

• Exercises: There will be exercises for this lecture. Participation is possible for those present in the first meeting at the beginning of October. In general, presence is mandatory for the exercise class. Students being absent more than once will be signed out. This will not apply in case of illness with a medical certificate. The rules may be adapted due to the present COVID-19 situation.
There are 38 exercises in total. Exercises 7,14,21,28,35 are extra tasks, which are voluntary. For each week everyone is supposed to prepare three exercises, i.e., we start with 1,2,3 for the second week, then 4,5,6 (and extra 7 if you like) for the third week, then 8,9,10, etc.

Exam 2022:

Monday, January 31-th, 13:15 - 14:45, Written exam by Zoom.

## Files:

 No. Thema Datum file 1 Syllabus 01.10.2021 Contents.pdf 2 My Lecture Notes 21.11.2022 Algebraic Number Theory 3 Milne's Lecture Notes 01.10.2021 Algebraic Number Theory 4 Exercises 01.10.2021 Exercises.pdf

Topics for the Exam:

1. Integral Ring Extensions
2. Ideals of Dedekind domains
3. The Class Number of Number Fields
4. Dirichlet's Unit Theorem
5. Decomposition and Ramification
6. Cyclotomic Fields
7. Valuations and Local Fields
8. Optional: Theorem of Kronecker-Weber

Dietrich Burde