• 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

1. Groups, Homomorphisms, Examples
2. Subgroups and cosets
3. Quotient of groups
4. Group operations
5. p-Groups and Sylow subgroups
6. Examples for the classification of groups
7. Simple and solvable groups
8. Rings and ideals
9. Integrity rings and prime ideals
10. Factorial rings
11. Division with remainder and Euclidean rings
12. Divisibility in polynomial rings
13. Moduli
14. Structure of moduli over principal ideal rings
15. Field extensions
16. Construction with compass and ruler
17. Algebraic closure
18. Separability
19. Main Theorem of Galois Theory
20. Examples for the calculation of Galois groups
21. Unity roots
22. Cyclic expansions
23. Solvability of equations