Math 6140 Spring 25

MATH 6140: Algebra 2 (Spring 2025)

Syllabus (see also the Algebra Prelim Syllabus)

Office hours:

Monday 10-11 am in Math 310,
Tuesday 4 - 5 pm in Math 310, and
by appointment

Exams:

Schedule

Numbers are for orientation and refer to sections with related material in Dummit-Foote: Abstract Algebra.
  1. 01/13: rings, modules (10.1) [notes]
  2. 01/15: R-algebras, submodules, homomorphisms (10.2) [notes]
  3. 01/17: quotients, sums, isomorphism theorems, generation of modules (10.3) [notes]
  4. 01/22: (direct) sums, free modules, universal mapping property (10.3), [notes]
  5. 01/24: tensor products, extending scalars (10.4) [notes]
  6. 01/27: tensor products of bimodules, bilinear maps (10.4) [notes]
  7. 01/29: vector spaces, bases (11.1) [notes]
  8. 01/31: matrix of linear transformations, similarity (11.2) [notes]
  9. 02/03: dual space, linear functionals (11.3) [notes] (add-on [infinite dimensional dual space])
  10. 02/05: determinants (11.4) [notes]
  11. 02/07: tensor algebras (11.5) [notes]
  12. 02/10: symmetric and exterior algebras (11.5)
  13. 02/12: Noetherian modules (12.1) [notes]
  14. 02/14: finite presentations of modules
  15. 02/17: Fundamental Theorem of finitely generated modules over PIDs, invariant factors, elementary factors (12.1) (add-on [rank of modules over integral domain])
  16. 02/19: characteristic and minimal polynomial, companion matrix, rational canonical form (12.2) [notes]
  17. 02/21: Jordan canonical form (12.3) [notes]
  18. 02/24: field extensions, characteristic, prime subfield (13.1) [notes]
  19. 02/26: polynomial and simple extensions (13.1)
  20. 02/28: algebraic extensions, minimal polynomial, Lagrange's Thm (13.2) [notes]
  21. 03/03: finitely generated algebraic extensions, algebraic subfields, composite fields (13.2)
  22. 03/05 splitting fields, existence and uniqueness (13.4) [notes]
  23. 03/07: straightedge and compass constructions (13.3) [notes]
  24. 03/10: algebraically closed fields, existence and uniqueness of algebraic closure (13.4)
  25. 03/12: separable vs irreducible polynomials (13.5) [notes]
  26. 03/14: finite fields, Frobenius endomorphism, perfect fields (13.5)

Assignments

Numbers refer to problems in Dummit-Foote: Abstract Algebra.
  1. (due 01/22) 10.1: 4, 9, 11, 21; 10.2: 3, 5, 7, 10
  2. (due 01/29) 10.3: 10, 11, 18, 20
  3. (due 02/05) 10.4: 2,4, 10, 14; 11.1: 6, 10, 13; 11.2: 8, 10, 11
  4. (due 02/12) 11.3; 2bd, 4, 5; 11.4: 4,6; 11.5: 1, 5 (cf. Proof of Cor 37), 13
  5. (due 02/19) 12.1: 2,5,9,11
  6. (due 02/26) 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 (no need to type matrix calculations, final results suffice for computational problems)
  7. (due 03/05) 12.3: 48,49; 13.1: 1, 3; 13.2: 1, 3, 7, 10, 19, 20
  8. (due 03/12) 13.3: 2,3,4 (no need to latex figures, refer to figures in the book or scan handdrawn sketches); 13.4: 4, 5, 6; [pdf]
  9. (due 03/19) 13.5: 3,4,6,7, 13.6: 1,5,8,9,11,12