# Number Theory II

Für eine deutsche Version dieser Ankündigung, klicken Sie bitte hier.

## Generalities

Lectures: Lars Kindler
Mondays, 10am-2pm (with a break), SR005, Arnimallee 3

Exercises: Lei Zhang
Wednesdays, 10am-12pm, SR005/A3, Arnimallee 3

Official website in course registry

or come to office 112A, Arnimallee 3

Prerequisites: Lecture "Algebra I" or roughly equivalent knowledge, i.e. familiarity with basic notions of algebra and commutative algebra.

## Description

This course gives an introduction to algebraic number theory. The main objects of study are number fields, i.e. finite extensions of the field of rational numbers. To a number field $K$ we will attach its ring of integers $\mathcal{O}_K$. The ring $\mathcal{O}_K$ is a Dedekind domain and we will see that one of its invariants is the class number $h_K$, which measures "how far" $\mathcal{O}_K$ is away from being a unique factorization domain.
We will also study finite extensions $K\subset L$ of number fields, and how the prime ideals behave in the associated extension $\mathcal{O}_K\subset \mathcal{O}_L$ of Dedekind domains.

Here is a rough outline of the course (subject to change):
1. Rings of integers
2. Basic properties of Dedekind domains
3. Minkowski's theory and finiteness of the class number
4. Dirichlet's Unit Theorem
5. Extensions of Dedekind domains and ramification theory
6. Absolute values and completions, local fields

## Literature

This course will mainly follow:
• Jürgen Neukirch: Algebraische Zahlentheorie, Springer Verlag (English translation also available)
• James Milne: Algebraic Number Theory (freely available here )