Syllabus Math 200B

1. Basic ring theory. Definitions examples. 7.1,7.2. Semigroup algebras and free algebras.

2. Ring Homorphisms. 7.3,7.4

3. Basic module theory. 10.1.

4. Module homorphisms 10.2,10.3

5. Euclidean domains 8.1

6. Principal ideal domains (PID) and unique factorization 8.2, 8.3

7. Fundamental theorem for finitely generated modules over a PID(invariant factor form) 12.1, pp.438-443.

8. Chinese remainder theorem 7.6.

9. Elementary divisors 12.1,pp. 444-451.

10. The fundamental theorem for finitely generated abelian groups, 5.2.

11. Basic vector space theory, 11.1,2.

12. Determinants over commutative rings with identity, 11.4

13. Cayley-Hamilton theorem, 12.2.

14. The Jordan canonical form, 12.3

15. Field extensions, 13.1

16. Algebraic extensions, 13.2

17. Splitting fields, 13.4

18. The fundamental theorem of algebra and the first fundamental theorem for S_n.

19. Algebraically closed fields and algebraic closure, 13.4

20. The isomorphism of algebraic closures.

21. Separable and inseparable extensions, 13.5

22. The classification of finite fields. 13.5

23. Cyclotomic extensions of the rational numbers, 13.6