## 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**
Please send questions and comments to kindler - at - math.fu-berlin.de

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):

- Rings of integers
- Basic properties of Dedekind domains
- Minkowski's theory and finiteness of the class number
- Dirichlet's Unit Theorem
- Extensions of Dedekind domains and ramification theory
- Absolute values and completions, local fields