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.