MTHSC 851, Spring 2009 / MTHSC 852, Fall 2009

MTHSC 851, Spring 2009 / MTHSC 852, Fall 2009


"You see this little hole? This moth's just about to emerge. It's in there right now, struggling. It's digging it's way through the thick hide of the cocoon. Now, I could help it - take my knife, gently widen the opening, and the moth would be free - but it would be too weak to survive. Struggle is nature's way of strengthening it."
--Locke (Lost, 2004)
Instructor: Dr. Matthew Macauley
Class: MthSc 851/852, Modern Algebra
MthSc 851 Class Syllabus
MthSc 852 Class Syllabus

Lecture Notes

Part I: Groups

Groups, Part 1: Groups, subgroups, and homomorphisms. (15 pages. Updated Feb 3, 2009)
Groups, Part 2: Permutation groups and group actions. (6 pages. Updated Feb 3, 2009)
Groups, Part 3: The symmetric and alternating groups. (7 pages. Updated Feb 3, 2009)
Groups, Part 4: The Sylow theorems. (9 pages. Updated Feb 3, 2009)
Groups, Part 5: Universal properties, solvable groups, and (sub)-normal series. (10 pages. Updated Apr 7, 2009)
Groups, Part 6: Categories, products, and coproducts. (14 pages. Updated Apr 7, 2009)
Groups, Part 7: Nilpotent groups and finite abelian groups. (8 pages. Updated Mar 3, 2009)
Groups, Part 8: Free groups, free objects, and group presentations. (16 pages. Updated Feb 25, 2010)

Part II: Rings

Rings, Part 1: Rings, ideals, homomorphisms, and fraction fields. (11 pages. Updated Apr 21, 2009)
Rings, Part 2: Polynomial rings. (10 pages. Updated Aug 20, 2009)
Rings, Part 3: Divisibility and factorization. (17 pages. Updated Aug 27, 2009)
Rings, Part 4: The Chinese remainder theorem. (4 pages. Updated Sept 3, 2009)
Rings, Part 5: The Hilbert basis theorem. (6 pages. Updated Sept 3, 2009)

Part III: Fields

Fields, Part 1: Field extensions. (12 pages. Updated Sept 17, 2009)
Fields, Part 2: Fundamental theorem of Galois theory. (12 pages. Updated Nov 6, 2009)
Fields, Part 3: Normal and separable extensions. (12 pages. Updated Oct 22, 2009)
Fields, Part 4: Galois theory of polynomials. (13 pages. Updated Oct 22, 2009)
Fields, Part 5: Transcendental field extensions. (11 pages. Updated Oct 27, 2009)

Part IV: Modules

Modules, Part 1: Preliminaries. (9 pages. Updated Nov 4, 2009)
Modules, Part 2: Direct sums and free modules. (11 pages. Updated Nov 4, 2009)
Modules, Part 3: Projective and injective modules. (10 pages. Updated Nov 11, 2009)
Modules, Part 4: Tensor products. (14 pages. Updated Dec 1, 2009)
Modules, Part 5: Modules over a PID. (14 pages. Updated Nov 23, 2009)
Modules, Part 6: Applications to linear algebra. (10 pages. Updated Dec 3, 2009)

Homework

Homework 1. Due Wednesday, January 21th 2009.
Homework 2. Due Wednesday, January 28th 2009.
Homework 3. Due Friday, February 6th 2009.
Homework 4. Due Friday, February 13th 2009.
Homework 5. Due Monday, February 23th 2009.
Homework 6. Due Tuesday, March 3rd 2009.
Homework 7. Due Thursday, March 26th 2009.
Homework 8. Due Thursday, April 2nd 2009.
Homework 9. Due Tuesday, April 14th 2009.
Homework 10. Due Thursday, April 24th 2009.

Homework 11. Due Friday, August 28th 2009.
Homework 12. Due Monday, September 7th 2009.
Homework 13. Due Monday, September 21st 2009.
Homework 14. Due Friday, October 9th 2009.
Homework 15. Due Monday, October 18th 2009.
Homework 16. Due Monday, November 2nd 2009.
Homework 17. Due Monday, November 16th 2009.
Homework 18. Due Monday, December 6th 2009.

Exams

MthSc 851, Midterm 1
MthSc 851, Midterm 2
MthSc 851, Final exam

MthSc 852, Midterm 1
MthSc 852, Midterm 2
MthSc 852, Final exam