Math 6140 Spring 22

MATH 6140: Algebra 2 (Spring 2022)

Syllabus (see also the Algebra Prelim Syllabus)

Office hours:

Monday 10-11 am in Math 310, Tuesday 2:30-3:20 pm on Zoom, and by appointment

Exams:

Schedule

Numbers are for orientation and refer to sections with related material in Dummit-Foote: Abstract Algebra.
  1. 01/10: rings, modules (10.1) [notes]
  2. 01/12: R-algebras, submodules, homomorphisms (10.2) [notes]
  3. 01/14: quotients, sums, isomorphism theorems, generation of modules (10.3) [notes]
  4. 01/19: (direct) sums, free modules, universal mapping property (10.3), [notes]
  5. 01/21: tensor products, extending scalars (10.4)
  6. 01/24: tensor products, bimodules, bilinear maps (10.4) [notes]
  7. 01/26: vector spaces, bases (11.1) [notes]
  8. 01/28: matrix of linear transformations, similarity (11.2) [notes]
  9. 01/31: dual space, linear functionals (11.3) [notes], [infinite dimensional dual space]
  10. 02/04: determinants (11.4) [notes]
  11. 02/07: tensor algebras, symmetric and exterior algebras (11.5) [notes]
  12. 02/09: Noetherian modules (12.1) [notes]
  13. 02/11: finite presentations of modules
  14. 02/14: Fundamental Theorem of finitely generated modules over PIDs, invariant factors, elementary factors (12.1)
  15. 02/16: characteristic and minimal polynomial, companion matrix, rational canonical form (12.2) [notes]
  16. 02/16: Jordan canonical form (12.3) [notes]
  17. 02/19: field extensions, characteristic, prime subfield, polynomial and simple extensions (13.1) [notes]
  18. 02/23: algebraic extensions, minimal polynomial, Lagrange's Theorem (13.2) [notes]
  19. 02/25: finitely generated algebraic extensions (13.2), straightedge and compass constructions (13.3)) [notes]
  20. 02/28: splitting fields, existence and uniqueness (13.4) [notes]
  21. 03/02: cyclotomic field, algebraically closed fields, existence and uniqueness of algebraic closure (13.4)
  22. 03/04: separable vs irreducible polynomials (13.5) [notes]
  23. 03/07: finite fields, Frobenius endomorphism, perfect fields (13.5)
  24. 03/09: cyclotomic fields (13.6) [notes]
  25. 03/11: field automorphisms Aut(K/F), action on roots (14.1) [notes]
  26. 03/14: fixed field Fix(G), Galois closures, normal, separable, Galois extension (14.1)
  27. 03/16: characters of groups, independence of field automorphisms (14.2) [notes]
  28. 03/18: Fundamental Theorem of Galois Theory (14.2)
  29. 03/28: Galois group of polynomials (14.2)
  30. 03/30: Galois group and algebraic closure of finite fields (14.3) [notes]
  31. 04/01: composite extensions, (sub)direct product of Galois groups (14.4) [notes]
  32. 04/04: Artin's Primitive Element Theorem (14.4)
  33. 04/06: discussion of 2nd midterm
  34. 04/08: inverse Galois problem, cyclotomic, abelian extensions, construction of n-gons (14.5) [notes]
  35. 04/11: symmetric functions, S_n as Galois group, discriminant (14.6) [notes]
  36. 04/13: Galois groups of polynomials, Fundamental Theorem of Algebra (14.6)
  37. 04/15: Kummer's theory on radical extensions, root extensions (14.7) [notes]
  38. 04/18: Galois' Theorem for solvability of a polynomial, unsolvability of quintics (14.7), Galois groups over Q and F_p (14.8) [notes]
  39. 04/20: algebraic independence, transcendence basis (14.9) [notes]
  40. 04/22: transcendental extensions, infinite Galois extensions (14.9)
  41. 04/25: review [notes]
  42. 04/27: review

Assignments

Numbers refer to problems in Dummit-Foote: Abstract Algebra.
  1. (due 01/19) 10.1: 4, 9, 11, 21; 10.2: 3, 5, 7, 10
  2. (due 01/26) 10.3: 10, 11, 18, 20
  3. (due 02/02) 10.4: 2,4, 10, 14; 11.1: 6, 10, 13; 11.2: 8, 10, 11
  4. (due 02/09) 11.3; 2bd, 4, 5; 11.4: 4,6;
  5. (due 02/16) 11.5: 1, 5 (cf. Proof of Cor 37), 13; [pdf], 12.1: 2,5,9,11;
  6. (due 02/23) 12.2: 6 (use that the characteristic polynomial splits into linear factors for the second half),8,9,10; 12.3: 9, 11, 17, 22, 31, 32, 48, 49 (no need to type matrix calculations, final results suffice for computational problems)
  7. (due 03/02) 13.1: 1, 3; 13.2: 1, 3, 7, 10, 19, 20; 13.3: 2,3,4 (no need to latex figures, refer to figures in the book or scan handdrawn sketches)
  8. (due 03/09) 13.4: 4, 5, 6; [pdf]; 13.5: 3,4,6
  9. (due 03/16) 13.5: 7, 13.6: 1,5,6,8ab,8cd,9,11,12
  10. (due 03/30) 14.1: 4,5,7,8, [pdf]; 14.2: 1,8,20
  11. (due 04/06) 14.2 4,5,10,14,29; 14.3: 4,5; 14.4: 1
  12. (due 04/13) 14.4: 2; 14.5: 5, 7, 10, 11; 14.6: 25, 46
  13. (due 04/20) 14.6: 16, 22 ; 14.7: 1,3,13,20; 14.8: 3 (use 14.6.35 without proof)