Math 418: Abstract Algebra II

Course Information

Illini Hall 7, TR 9:30-10:50am.

Instructor: Rebecca Tramel.

Office Hours: Illini Hall 241 Tuesdays 11-12, Wednesday 3-4, or by appointment.

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

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 11): 25%

Midterm Exams (Feb. 16, Mar. 17): 35%

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 two midterm exams are scheduled for Tuesday, February 16 and Thursday, March 17. 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. 19, 21): Review of rings, Sections 8.1 and 8.2 on Euclidean Domains and Principal Ideal Domains.- HW1 Due Jan 26: 8.1 #3, 7 (only the pair 85, 1+13i),
8a (only D=-2, -3), 10. Solutions.

**Week 2**(Jan 26, 28): Sections 8.2 and 8.3 on Principal Ideal Domains and Unique Factorization Domains.- HW2 Due Feb 4: 8.2 #1, 5, 8.3 #8, 9. Solutions.

- Worksheet 1 (Jan 26) and Solutions.
**Week 3**(Feb 2, 4): Sections 9.1, 9.2, and 9.3 on Polynomial Rings over Fields and UFDs.- HW3 Due Feb 11: 9.1 #4, 9.2 #1,5 (Hint: The 4th Isomorphism Theorem for Rings may help.), 9.3 #4 (for part d, just explain what the zero-divisors and units in this quotient ring are.) Solutions.
**Week 4**(Feb 9, 11): Sections 9.4, 9.5, and 9.6 on Irreducible Polynomials and Polynomial Rings in Several Variables.- Suggested Exercises (to practice for the exam): 9.4 #1, 2,
14, 9.6 #10, 11, 12, 40. Solutions.

**Week 5**(Feb 16, 18): Midterm 1, Section 13.1 on Field Extensions.- Midterm Exam 1 is in class on Tuesday, February 16. There is
a take home
portion of the exam, due in class on Tuesday. You may use your
book and notes, but no other sources for the take home
portion. The in class exam is closed book. Solutions.

- HW 4 Due Feb 23: 13.1 #2, 3, 4. Solutions.
**Week 6**(Feb 23, 25): Section 13.2 and 13.3 on Algebraic Extensions and Straightedge and Compass Constructions.- HW 5 Due Mar 1: 13.2 #3, 7, 19, 20 (Hint: Look at p.473 to
review characteristic polynomials. What is a root of the
characteristic polynomial? It may also help to recall the
construction of the matrix of a linear transformation on p.
415-416). Solutions.

**Week 7**(Mar 1, 3): Section 13.4 and 13.5 on Splitting Fields and Separable Extensions.- HW 6 Due Mar 8: 13.4 #1, 3, 5, 13.5 #7. Solutions.

**Week 8**(Mar 8, 10): Section 14.1 and 14.2 on Galois Theory.- HW 7 Due Mar 15: 14.1 #1, 5, 7, 14.2 #6. Solutions.

**Week 9**(Mar 15, 17): Midterm 2 and 14.2 on Galois Theory.- Midterm 2 is in class on Thursday, March 17. There is a
review sheet for this exam here. Some
suggested practice problems are: 13.2 #13, 13.4 #2, 13.6
#10, 14.1 #10, 14.2 #11, 13. The take home portion of the exam
is here, it is
due Tuesday, March 29. Solutions.

**Week 10**(Mar 29, 31): Sections 10.1 and 10.2 on Modules.- HW 8 Due Apr 5: 10.1 #4, 8, 9, 11. (For 11, you will also need to use the definition in exercise 10, which is similar to exercise 9.) Solutions.
**Week 11**(Apr 5, 7): Sections 10.2, and 10.3 on Modules.

- HW 9 Due Apr 14: 10.2 #8, 9, 10.3 #12, 15. Solutions.

**Week 12**(Apr 12, 14): Section 12.1 on Modules over PIDs.- HW 10 Due Apr 21: 12.1 #5, 6, 17, 18, 19.
**Week 13**(Apr 19, 21): Sections 12.2 and 12.3 on the Rational Canonical Form and the Jordan Canonical Form.- HW 11 Due Apr 28: 12.3 #11, 18. Handout
(Atiyah-MacDonald) #15, 17i-iv. (Don't worry about showing
that the sets are a basis for the topology, just do parts
i-iv.). Solutions.

**Week 14**(Apr 26, 28): Algebraic Geometry- Here are the
notes from class so far. They will be regularly updated based
on what we've covered. These notes take from a variety of
sources, such as Hartshorne's Algebraic Geometry, Atiyah and
MacDonald's Introduction to Commutative Algebra, Gathmann's
online Algebraic Geometry notes, and our text book.

**Week 15**(May 3): Algebraic Geometry and Review- We will have a
**review session**on Thursday at 11am in Altgeld 159. Some suggested exercises are: - Chapter 8: 8.1 #9, 11, 8.2 #3, 7, 8.3 #5.
- Chapter 9: 9.2 #6, 9.3 #1, 9.4 #1, 2, 6, 9.6 #2, 4, 14, 15.
- Chapter 13: 13.1 #5, 13.2 #8, 11, 17, 13.4 #3, 13.5 #5, 11.
- Chapter 14: 14.1 #4, 6, 14.2 #7, 9.
- Chapter 10: 10.1 #14, 19, 10.2 #3, 6, 10.3 #7, 9, 22.
- Chapter 12: 12.1 #9, 11, 13, 16, 12.2 #14, 12.3 #10.
- Algebraic Geometry: Exercises.
Solutions.

**Final Exam**(May 12): Thurday May 12 at 7pm in Illini 7. Here is a study guide for the exam.

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).