Math 6140 Spring 18

MATH 6140: Algebra 2 (Spring 2018)

Syllabus (see also the Algebra Prelim Syllabus)

Office hours:

Monday 4-5 pm, Tuesday 3-4 pm, and by appointment

Schedule

Numbers are for orientation and refer to sections with related material in Dummit-Foote: Abstract Algebra.
  1. 01/17: rings, modules (10.1)
  2. 01/19: R-algebras, submodules, homomorphisms (10.2)
  3. 01/22: isomorphism theorems
  4. 01/24: generation of modules, (direct) sums (10.3), free modules
  5. 01/26: vector spaces (11.1)
  6. 01/29: universal property of modules
  7. 01/31: matrix of linear transformations, similarity (11.2)
  8. 02/02: dual space, linear functionals (11.3)
  9. 02/05: determinants (11.4)
  10. 02/07: Noetherian modules (12.1)
  11. 02/09: Fundamental Theorem of finitely generated modules over PIDs, invariant factors, elementary factors (12.1)
  12. 02/12: characteristic and minimal polynomial, companion matrix, rational canonical form (12.2)
  13. 02/14: Fundamental Theorem of fg modules over PIDs (uniqueness) (12.1)
  14. 02/16: Jordan canonical form (12.3)
  15. 02/19: field extensions, characteristic, prime subfield (13.1)
  16. 02/21: polynomial vs simple extensions (13.1)
  17. 02/23: algebraic extensions, minimal polynomial, Lagrange's Theorem (13.2)
  18. 02/26: finitely generated algebraic extensions (13.2)
  19. 02/28: straightedge and compass constructions (13.3)
  20. 03/02: splitting fields, existence and uniqueness (13.4)
  21. 03/05: cyclotomic field, algebraically closed fields (13.4)
  22. 03/07: existence and uniqueness of algebraic closure (13.4)
  23. 03/09: separable vs irreducible polynomials (13.5)
  24. 03/12: finite fields, Frobenius endomorphism, perfect fields (13.5)
  25. 03/14: cyclotomic fields (13.6)
  26. 03/16: field automorphisms, Aut(K/F), action on roots (14.1)
  27. 03/19: fixed field Fix(G), Galois closures (14.1)
  28. 03/21: normal, separable, Galois extension (14.1)
  29. 03/23: characters of groups, independence of field automorphisms (14.2)
  30. 04/02: Fundamental Theorem of Galois Theory (14.2)
  31. 04/04: Galois group of polynomials (14.2)
  32. 04/06: Galois group and algebraic closure of finite fields (14.3)
  33. 04/09: composite extensions, (sub)direct product of Galois groups (14.4)
  34. 04/11: discussion of 2nd midterm
  35. 04/13: Artin's Primitive Element Theorem (14.4)
  36. 04/16: inverse Galois problem, cyclotomic, abelian extensions, construction of n-gons (14.5)
  37. 04/18: symmetric functions, S_n as Galois group, discriminant (14.6)
  38. 04/20: Galois groups of polynomials, Fundamental Theorem of Algebra (14.6)
  39. 04/23: Kummer's theory on radical extensions (14.7)
  40. 04/25: root extensions (14.7)
  41. 04/27: Galois' Theorem for solvability of a polynomial (14.7)
  42. 04/30: unsolvability of quintics (14.7)
  43. 05/01: transcendent extensions (14.9)

Assignments

Numbers refer to problems in Dummit-Foote: Abstract Algebra.
  1. (due 01/24) 10.1: 4, 9, 11, 21; 10.2: 3, 5, 7, 10
  2. (due 01/31) 10.3: 10, 11, 18; 11.1 6, 10, 13; [pdf]
  3. (due 02/07) 11.2: 8. 10, 11; 11.3; 2bd, 4, 5; 11.4: 4,6
  4. (due 02/14) 12.1: 2,5,9,11; 12.2: 6,8,9,10
  5. (due 02/21) 12.3: 9, 11, 17, 22, 31, 32, 48, 49
  6. (due 02/28) 13.1: 1, 3; 13.2: 1, 3, 7, 10, 19, 20
  7. (due 03/07) 13.3: 4; 13.4: 4, 5, 6; [pdf]
  8. (due 03/14) 13.5: 3,4,6,7
  9. (due 03/21) 13.6: 1,5,6,8ab,8cd,9,11,12
  10. (due 04/04) 14.1: 4,5,7,8,10; [pdf]
  11. (due 04/11) 14.2: 1,4,5,8,10,14,20,29
  12. (due 04/18) 14.3: 4,5; 14.4: 1, 2; 14.5: 5, 7, 10, 11
  13. (due 04/25) 14.6: 8, 16, 25, 46; 14.7: 3,13,20; 14.8: 3