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.


  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