Math 6140 Spring 17

MATH 6140: Algebra 2 (Spring 2017)

Syllabus

Office hours:

Monday 4-5 pm, Tuesday 4-5 pm, and by appointment

Schedule

Numbers are for orientation and refer to sections with related material in Dummit-Foote: Abstract Algebra.
  1. 01/18: rings, modules (10.1)
  2. 01/20: submodules, homomorphisms, endomorphism ring (10.2)
  3. 01/23: quotients, isomorphism theorems, generation of modules (10.3)
  4. 01/25: internal and external direct sums (10.3)
  5. 01/27: basis, free module (10.3), vector spaces (11.1)
  6. 01/30: replacement theorem, dimension (11.1)
  7. 02/01: universal property of free modules (10.3), matrix of linear transformation, similarity (11.2)
  8. 02/03: linear functionals, dual spaces (11.3)
  9. 02/06: determinants (11.4)
  10. 02/08: Noetherian modules (12.1)
  11. 02/10: Fundamental Theorem of finitely generated modules over PIDs, invariant factors (12.1)
  12. 02/13: Fundamental Theorem, elementary factors, uniqueness (12.1)
  13. 02/15: characteristic and minimal polynomial, companion matrix (12.2)
  14. 02/17: rational canonical form (12.2)
  15. 02/20: Jordan canonical form (12.3)
  16. 02/22: characteristic, prime field, polynomial extension F[x]/(p(x) (13.1)
  17. 02/24: simple extensions (13.1)
  18. 02/27: algebraic extension, minimal polynomial, degree (13.2)
  19. 03/01: finitely generated extensions (13.2)
  20. 03/03: splitting fields, existence and uniqueness (13.4)
  21. 03/06: disussing 1st midterm, algebraic closure (13.4)
  22. 03/08: algebraic closure, existence and uniqueness (13.4)
  23. 03/10: doubling cube, trisecting angles, squaring circle with straightedge and compass (13.3)
  24. 03/13: size of Aut(K/F) (14.1)
  25. 03/15: separable polynomials, Frobenius endomorphism, finite fields F_q (13.5)
  26. 03/17: separable extensions, perfect fields (13.5)
  27. 03/20: cyclotomic fields (13.6)
  28. 03/22: field automorphisms, Aut(K/F), action on roots (14.1)
  29. 03/24: fixed field Fix(G), Galois closures (14.1)
  30. 04/03: normal, separable, Galois extension (14.1)
  31. 04/05: characters, independence of field automorphisms (14.2)
  32. 04/07: Fundamental Theorem of Galois Theory (14.2)
  33. 04/10: Galois group of polynomials (14.2)
  34. 04/12: discussing 2nd midterm,
  35. 04/14: Galois group, algebraic closure of finite fields (14.3)
  36. 04/17: composite extensions, subdirect product of Galois groups (14.4)
  37. 04/19: simple extensions (14.4)
  38. 04/21: Artin's Primitive Element Theorem (14.4)
  39. 04/24: inverse Galois problem, cyclotomic, abelian extensions (14.5)
  40. 04/26: construction of n-gons (14.5), symmetric functions (14.6)
  41. 04/28: discriminant, Galois groups of polynomials (14.6)
  42. 05/01: Kummer's theory on radical extensions, root extensions (14.7)
  43. 05/03: unsolvability of quintics (14.7)
  44. 05/05: transcendent extensions (14.9)

Assignments

  1. (due 01/25) 10.1: 4, 9, 11, 21; 10.2: 3, 5, 7, 10
  2. (due 02/01): 10.3: 8, 10, 11, 18; 11.1: 6, 9, 10, 13
  3. (due 02/13): 11.2: 3, 8, 10, 34, 37ab; 11.3: 2ab, 4; 11.4: 4
  4. (due 02/20): 12.1: 2,5,9,11; 12.2: 6,8,9,10
  5. (due 02/24): 12.3: 9, 11, 17, 22, 31, 32, 48, 49
  6. (due 03/01): 13.1: 1, 3; 13.2: 1, 3, 7, 10, 19, 20
  7. (due 03/08): 13.4: 4, 5, 6
  8. (due 03/15): 13.3: 4; 13.5: 3,4,6
  9. (due 03/22): 13.5: 7, 13.6: 1,6,8ab,8cd,9,11,12
  10. (due 04/05): 14.1: 4, 5, 7, 8
  11. (due 04/12): 14.2: 1, 4, 5, 10
  12. (due 04/19): 14.2: 8,14,17,20,29; 14.3: 4,5,13
  13. (due 04/26): 14.3: 14, 15, 16, 17; 14.4: 1, 2; 14.5: 7, 10
  14. (due 05/03): 14.6: 16, 25, 46; 14.7: 3,20; 14.8: 3