# Algebra I

Undergraduate Course, *Ruhr University Bochum*, 2019

- Lecturer: Prof. Dr. Röhrle
- Language: German
- Credits: 9 CP
- Programs: B.A./B.Sc./M.Sc. Mathematics
- Examination: 100 % Written Exam (120 Minutes)

# Course Description

The lecture will give a systematic introduction to the theory of groups, rings, and fields and present some of the classical applications of this theory. Specifically, the following topics will be covered:

- Group theory: isomorphism theorems, permutation groups, group actions, resolvable and simple groups, Sylow theorms
- Ring theory: integrity rings, prinicpal ideal domains, prime factorization in rings and polynomial rings, module theory
- Field theory: minimal polynomial, algebraic extensions, separable and normal field extensions, Galois groups, and main theorem of Galois theory

In addition, some classical applications of Galois theory are discussed.

# Contents

- Groups, Homomorphisms, Examples
- Subgroups and cosets
- Quotient of groups
- Group operations
- p-Groups and Sylow subgroups
- Examples for the classification of groups
- Simple and solvable groups
- Rings and ideals
- Integrity rings and prime ideals
- Factorial rings
- Division with remainder and Euclidean rings
- Divisibility in polynomial rings
- Moduli
- Structure of moduli over principal ideal rings
- Field extensions
- Construction with compass and ruler
- Algebraic closure
- Separability
- Main Theorem of Galois Theory
- Examples for the calculation of Galois groups
- Unity roots
- Cyclic expansions
- Solvability of equations