Syllabus

Tentative Syllabus

1. Review of integers: gcd, lmc, Modular Arithmetic.

2. Equivalence Relations

3. Symmetries in Geometry and beyond

4. Groups

5. Basic properties of groups

6. Subgroups

7. Center and Centralizer

8. Cyclic groups

9. Fundamental Theorem for Cyclic groups

10. Permutation groups

11. Basic properties of permutations

12. Alternating groups

13. Isomorphisms

14. Properties of isomorphisms

15. Automorphisms

16. Cosets

17. Midterm I (?)

18. Lagrange's Theorem

19. Applications

20. External direct products

21. Applications

22. Normal subgroups

23. Internal direct products

24. Group homomorphism

25. Isomorphism theorems

26. Abelian groups

27. Rings

28. Integral domains

29. Midterm II (?)

30. Characteristic of a ring

31. Ideals

32. Factor rings

33. Ring homomorphism

34. Field of quotients

35. Polynomial rings

36. PIDs

37. Factorization of polynomials

38. UFDs

39. Fields

40. Field extensions

41. Finite fields

42. Finite fields

This is a more detailed list of topics that we discussed: