Math 8000, Fall 2010
Daniel Krashen
- Office hours: Tuesdays, 10-11am and 1-2pm in Boyd 437
- Problem session: Wednesday, 12-1pm in Boyd 410
Exam schedule:
- Midterm Exam
Thursday, October 7
- Final Exam
Tuesday, December 14, 12-3
General Departmental Resources
Supplimentary Materials
Homework Assignments
- Due Tuesday, August 24
- Textbook problems 0.1/7, 0.2/1(f), 0.3/4, 1.1/9,
12, 18, 21, 25
- Due Tuesday, August 31
- Textbook problems 2.1/8, 10, 13; 2.2/6, 7, 8,
13
- Determine the orbits of the dihedral group
D8 acting on itself via
conjugation.
- Due Tuesday, September 7
- Textbook problems 2.3/13, 19; 2.4/7, 14; 3.1/1, 22, 27, 35, 36
- Ungraded problems: 2.3/23; 3.1/3, 24, 28
- Due Tuesday, September 14
- Textbook problems 3.1/43; 3.2/12 (note the group is not
necessarily finite), 16, 18, 20; 3.4/8, 9, 10, 11; 4.1/10
- Due Thursday, September 23
- Due Thursday, September 30
- Due Thursday, October 7
- Textbook problems 7.1/7,15; 7.3/17,19
- Ungraded problems:
Practice Exam
(Correction: in problem 3(d), assume that A is finite!)
- Due Thursday, October 14
- Textbook problems 7.3/10, 13, 14, 24, 34; 7.4/ 8,
9, 10, 13 (Note -- all rings here are assumed to have an
identity!)
- Due Thursday, October 21
- Textbook problems 7.6/1, 6; 8.1/4; 8.2/3, 6, 8;
9.1/4; 9.2/4
- Due Thursday, October 28
- Textbook problems 10.1/5, 7, 13
- Ungraded problems: 10.1/2, 4, 6
- Due Thursday, November 4
- Textbook problems 10.4/2, 3, 4
- Read Chapter 11.1 (we'll go over the proof of
Theorem 3 in class on Tuesday).
- Textbook problems 11.1/2, 6, 8, 10
- Due Thursday, November 11
- Textbook problems 10.3/5, 6 (see definition, page
351), 7, 9, 10, 12, 13, 18, 19, 22
- Read Chapter 11.4
- Due Thursday, November 18
- Textbook problems 11.2/1, 2 (transition matricies
are discussed on page 419); 12.1/2, 3, 5, 7, 11,
13, 15
- Due Thursday, December 2
- Textbook problems 13.1/1 (see examples on pages
515, 518), 4; 13.2/1, 3, 5, 7, 10, 17, 19
Lecture Notes (scanned PDF):
- Equivalence Relations, Actions and Orbits (Lecture 2, August 19, 2010)
- Centralizers, Normalizers and Stabilizers (Lecture 3, August 24, 2010)
- Homomorphisms, Actions, Cosets, and Cyclic Groups (Lecture 4, August 26, 2010)
- Quotient Groups and the Fundamental Theorem
of Functions (Lecture 5, August 31, 2010)
- More about Normal Subgroups and the Isomorphism
Theorems (Lecture 6, September 2, 2010)
- The Correspondence Theorem and the Hölder Program (Lecture 7, September 7, 2010)
- Cosets, Actions and Permutations (Lecture 8, September 9, 2010)
- Permutations and Automorphisms; Intro to Sylow (Lecture 9, September 14, 2010)
- The Sylow Theorems (Lecture 10, September 16, 2010)
- Sylow Examples, Products and Semidirect Products (Lecture 11, September 21, 2010)
- Semidirect Products; Endomorphisms and Rings (Lecture 12, September 23, 2010)
- Group Presentations; Rings and Ideals (Lecture 13, September 28, 2010)
- Quotients and Isomorphism Theorems; Generated Ideals
(Lecture 14, September 30, 2010)
- Zorn's Lemma and Maximal Ideals; Prime Ideals;
Noetherian Rings; Products of Rings; Chinese Remainder
Theorem (Lecture 15, October 5, 2010)
- Fractions; Euclidean Domains and PIDs (Lecture 16, October 12, 2010)
- UFDs; Gauss' Lemma (Lecture 17, October 14, 2010)
- Polynomial Rings over UFDs are UFDs (Lecture 18, October 20, 2010)
- Modules (Lecture 19, October 21, 2010)
- Modules and Linear Transformations; Isomorphism
Theorems; Correspondence Theorems; Generation of Modules;
Products (Lecture 20, October 26, 2010)
- Tensor Products (Lecture 21, October 28, 2010)
- Vector Spaces (Lecture 22, November 2, 2010)
- Finitely generated modules over a PID, part 1 (Lecture 23, November 4, 2010)
- Finitely generated modules over a PID, part 2 (Lecture 24, November 9, 2010)
- Canonical Forms (Lecture 25, November 11, 2010)
- Minimal and Characteristic Polynomials (Lecture 26, November 16, 2010)
- Field Extensions (Lecture 27, November 18, 2010)
- Galois Theory (Lecture 28, November 30, 2010)
- Galois Theory, continued (Lecture 29, December 2, 2010)
