| Day | Topics | Preparation |
|---|---|---|
| Th. 1/21 | Introduction: Some classical problems. Constructible numbers, solution of x^3-2 . | [OS] Sec. 17, 18, 19. |
| Tu. 1/26 | Fundamental properties of polynomial rings over a field. Cubic field extensions of Q . |
[OS] Sec. 10-15, 19. |
| Th. 1/28 | Derivatives and irreducibility. Mason Stothers theorem and the abc conjecture. |
[A] Sec. 2.9 [OS] Sec. 16. |
| Tu. 2/2 | Mason Stothers theorem and the abc conjecture. Lattice of subgroups. Properties of the direct product. |
[OS] Secs 5. |
| Tu. 2/4 | Matrix groups. Automorphism groups. | [OS] Sec 10. |
| Tu. 2/9 | Symmetric and Alternating group. Simplicity of A_n |
[OS] Sec. 2 |
| Th. 2/11 | The second isomorphism theorem. Direct and semidirect products. Group Extensions |
[A] Sec. 1.4-5 |
| Tu. 2/16 | Classification of abelian groups. | [H] Sec. 8.2 |
| Th. 2/18 | Free groups and presentations. Cyclic extensions. |
[A] Sec 5.8. |
| Tu. 2/23 | No class: do problems | . |
| Th. 2/25 | TEST: Groups. | . |
| Tu. 3/1 | Orbit-stabilizer theorem | [A] 5.1,2 |
| Th. 3/3 | Online videos: Sylow Theorems | [A] 5.1-4. |
| Tu. 3/8 | Several problems on group actions. Groups of prime power order. | . |
| Th. 3/10 | Applications of Sylow theorems. A theorem about semi-direct products. Groups of order pq^r |
[A] 5.4,5. |
| Tu. 3/15 | Groups of order 27. Composition series for p -groups . |
[A] 5.5 and Groups of Small Order . |
| Th. 3/17 | Jordan-Holder theorem: uniqueness of factors in the
composition series. Solvability and nilpotency. . |
. |
| Tu. 3/22 | More on solvability, Sylow and group actions. | . |
| Th. 3/24 | TEST: Group Actions, Sylow theorems, group presentations, semi-direct products. | . |
| Tu. 4/5 | Finite Fields. | [OS] Sec 20. [A] Sec 6.4. |
| Th. 4/7 | Automorphisms of Finite Fields: The Frobenius map. | [OS] Sec 20 [A] Sec 6.4. |
| Tu. 4/12 | Separable and nonseparable extensions. The primitive element theorem. Field homomorphism. |
[A] Secs 3.1-4. [OS]. 22-25. |
| Th. 4/14 | Galois correspondence. | [OS] Sec 27. [A] 6.1-2. |
| Tu. 4/19 | Galois main theorem. | [OS] Sec 27. [A] 6.1-2. |
| Th. 4/21 | Cyclotomic fields extensions. | [A] 6.3,6. [A] 6.5 |
| Tu. 4/26 | Cyclic and abelian extensions. | [A] 6.7. |
| Tu 4/28 | Geometric constructions. | [A] 6.8. |
| Tu. 5/3 | Solvability by radicals | [A] 6.8 |
| Th. 5/5 | ||
| Tu. 5/10 | Final Exam | . |