Modern Algebra
Server-Angel Popescu

We will cover the following topics in this course.

1. Some arithmetic on N: Law of well ordering, Principle of mathematical induction, Division algorithm, Fundamental theorem of arithmetic, Fermat's little theorem and applications.

2. Equivalence relations, Order relations, the abstract construction of Z, Q and R.

3. The isomorphism theorem for sets.

4. Some axioms in Set Theory. Schroder-Berstein theorem.

5. Langrange's theorem and applications.

6. Direct products of groups and the first counting principle.

7. Isomorphism theorems for groups.

8. The Cauchy theorem for abelian groups.

9. The Sylow theorem for abelian groups.

10. The Cauchy theorem for non-abelian groups.

11. The Sylow theorem for non-abelian groups.

12. Cayley's theorem. Group actions. Orbits.

13. Conjugacy classes and counting principle.

14. Finite abelian p-groups: The structure theorem of finite abelian p-groups.

15. Free abelian groups. Exact sequences and the projective property of a free abelian group.

16. Finitely generated abelian groups. Torsion free finitely abelian generated groups. The free and the torsion part part of a finitely generated abelian group. The torsion subgroups.

17. Rings, prime ideals, maximal ideals, factor rings, the isomorphism theorems for rings.

18. Eclidean rings.

19. PIDs: irreducible and prime elements, prime ideals, maximal ideals.

20. The valuation rings K[[X]].

21. The Chinese remainder theorem for rings and ideals.

22. Noetherian rings. ascending chain condition. The Hilbert basis theorem for R[X].

23. Unique Factorization Domains.

24. Symmetric polynomials. The fundamental theorem of the symmetric polynomials.

25. Left modules, submodules, direct sums, direct products, free modules, exact sequences, projective and injective modules, Hom(M,N), isomorphism theorems for modules.

26. Tensor product of modules. Modules of fractions.