The course will give an introduction to basic concepts in field threory and ring theory.

The lectures will be in English.

In the **first week**, the lecture takes place on Wednesday **and Thursday** (9.45 to 11.15). The first problem class is on Friday, October 24, from 11.30 to 13.00.

In the **week before Christmas** the lecture and the tutorial will be swapped. The lecture will take place on **Tuesday, 16.12.2014, 17:30 in the room 1C-03** (Allianzgebäude) and the tutorial will take place on **Wednesday, 17.12.2014, 9:45 in the lecture hall AOC 201** (chemistry building).

There will be **no tutorial** at 23 December 2014.

**Contents**

**1. Field theory**

Algebraic field extensions, separability, normal extensions, Galois theory.

Roots of unity, cyclotomic fields, solution of polynomial equations by radicals, solvable groups.

**2. Valuation theory**

Valuations, absolute values and norms on rings.

*p*-adic numbers, nonarchimedean valuations.

Discrete valuation rings, extension of valuations.

**3. Ring theory**

Modules over rings, tensor product of modules, multilinear algebra.

Noetherian rings, ring of polynomials, Hilbert's basis theorem.

Integral ring extensions, integral closure; normalization, Dedekind domains.

# Exercise Sheets

The **exercise sheets** can be found in the Ilias course.

# Examination

The oral exams will take place on **2 March, 16 March and 30 March 2015**. Students who want to take the exam should **sign in to the time table** which can be found in the secretary's office (room 4A-21.1, Allianz building). Possible times are listed in the table there.

In addition you must register the exam in the **Campus Management for Students**.