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# Zahlentheorie II

### Allgemeine Informationen

Ort (Vorlesungen): SR 032/A6 Seminarraum (Arnimallee 6)

Ort (Übungen): SR 031/A6 Seminarraum (Arnimallee 6)

Zeit: Dienstag 10:15 - 11:45, 14:15-15:45 (Vorlesungen) und Donnerstag 16:15 - 17:45 (Übungen)

Erster Termin: 17.04.2017

Unterrichtssprache: English

### Beschreibung

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. This ring is a Dedekind domain and we will see that one of its invariants is the class number, which measures "how far" the ring is away from being a unique factorization domain. We will also study finite extensions of number fields, and how the prime ideals behave in the associated extensions of the rings of integers.

### Klausur

Zeit: Dienstag, 17. Juli, 10:00 (sharp) - 12:00 (sharp)

Oort: SR 032/A6

Kommentar: You may bring two sheets of paper (DIN A4), each written by yourself and by hand with writing on both sides

### Nachklausur

Kommentar: You may bring two sheets of paper (DIN A4), each written by yourself and by hand with writing on both sides

### Literatur

Die Vorlesung wird hauptsächlich folgenden Quellen folgen:

• Milne, James: Algebraic Number Theory (available here)
• Neukirch, Jürgen: Algebraische Zahlentheorie, Springer Verlag (English translation also available)

### Nötige Vorkenntnisse

Linear Algebra I+II, Algebra and Number Theory I

Here is a rough outline of the course (subject to changes):

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