Math 418: Abstract Algebra II

Course Information

Instructor: Rebecca Tramel.

Office Hours: Illini Hall 241 Tuesdays 11-12 and Wednesdays 12-1, or by appointment.

Text: Abstract Algebra (3rd Edition) by Dummit and Foote.

Meetings: Mechanical Engineering Building 336, Tuesdays and Thursdays 9:30-10:50am.

Topics

The goal of this course is to introduce important topics in the study of Algebra. The course will be divided into three major sections:

- Rings (Chapters 8 and 9),
- Fields (Chapters 13 and 14),
- Modules (Chapter 10 and 12).

Grading

Weekly Homework (Best 9 of 12): 25%

Midterm Exam (Mar. 2): 17.5%

Project: 17.5%

Final Exam: 40%

Homework assignments will be graded on a two-tier system:

- Most exercises will be graded out of 2 points, with full
credit awarded for a mostly correct solution.

- One exercise each week will be looked at in more detail, and will usually be graded out of 4 points. For this exercise, a perfect score will be reserved for a perfect solution.
- Another 2 points will be awarded to each assignment based on
its neatness and organization. In order to ensure you always
earn these points, make sure to staple your assignments, label
your exercises, and either type your solutions or write them out
neatly.

Schedule

The midterm exam is scheduled for March 2nd.. The final exam is scheduled by the Registrar's office, and so the date and time will be listed later in the semester. Keep these dates in mind while scheduling holiday travel.

**Week 1**(Jan. 17, 19): Review of rings and integral domains, Section 8.1 on Euclidean domains.- Jan. 17 Worksheet: Frequently
Cited Results, Filled
in version.

- HW1 Due Jan 26: 8.1 #3, 7, 8a (D=-2, -3 only), 8.2 #1. Solutions.

**Week 2**(Jan. 24, 26): Sections 8.2 and 8.3 on Principal Ideal domains and Unique Factorization domains.- HW2 Due Feb 2: 8.2 #5, 8.3 #5a,b, 8. Solutions.

**Week 3**(Jan. 31, Feb. 2): Sections 9.1 and 9.2 on polynomial rings over fields.- HW3 Due Feb 9: 9.1 #7, 9.2 #5, 6a,b,c (don't worry about
characteristic), 9.3 #3. Solutions.

**Week 4**(Feb. 7, 9): Sections 9.3 and 9.4 on irreducible polynomials.- HW4 Due Feb 16: 9.3 #2, 9.4 #2, 4, 9. Solutions.

**Week 5**(Feb. 14, 16): Sections 9.5 and 13.1 on polynomials in several variables and on field extensions.- HW5 Due Feb 23: 13.1 #2, 3, 13.2 #4, 7. Solutions.

**Week 6**(Feb. 21, 23): Section 13.2 and 13.4 on algebraic field extensions and splitting fields.- No HW Due next week!
- Practice Midterm.
Solutions.

**Week 7**(Feb. 28, Mar. 2): Section 13.5 on splitting fields, Midterm Exam.- Feb. 28 Review game (with solutions).
- HW6 Due Mar 9: 13.4 #1, 2,13.5 #5. Solutions.

- Midterm Exam. You can correct one page of your exam and
submit it March 9 to receive up to 75% of your lost points
back. Midterm with
solutions.

**Week 8**(Mar. 7, 9): Section 14.1 and 14.2 on Galois theory.- HW7 Due Mar 16: 14.1 #1, 2, 3. Solutions.

**Week 9**(Mar. 14, 16): Section 14.2 on Galois theory.- HW8 Due Mar 30: 14.2 #6. (Yes, it's just one problem, but
give yourself plenty of time!) Solutions.

**Week 10**(Mar. 28, 30): Sections 10.1 and 10.2 on modules.**Week 11**(Apr. 4, 6): Section 10.3 and 12.1 on modules.- HW10 Due Apr 13: 10.1 #21, 10.2 #8, 9, 10.3 #13. Solutions.

**Week 12**(Apr. 11, 13): Sections 12.2 and 12.3 on canonical forms of matrices.

- HW11 Due Apr 20: 12.1 #5, 12.2 #14, 12.3 #10. Solutions.

**Week 13**(Apr. 18, 20): Work on projects.- Class Apr. 18 Outside

**Week 14**(Apr. 25, 27): Group presentations.- Tuesday Apr. 25:
- 9:30 Representation Theory
- 9:50 Lie Groups
- 10:10 Cyclotomic Fields
- 10:30 Insolvability of the Quintic
- Thursday Apr. 27:
- 9:30 Algebraic Geometry

- 9:50 Banach Algebras

- 10:10 Groebner Bases

**Week 15**(May 2): Group presentations, review.- Practice Final
Questions. Solutions.

- The final exam will be cumulative, with a focus on Chapters
14, 10, and 12.

**Final Exam**: TBA.

Syllabus

LaTeX template - You will need to download a program to compile the document. There are many options, for example TeXStudio (Windows), TeXShop or MacTeX (Mac), and Kile (Linux).