Math 417: Abstract Algebra I

Course Information

Instructor: Rebecca Tramel.

Office Hours: Illini Hall 241 Monday 3-5 and Tuesday 1-2, or by appointment.

Text: Algebra: Abstract and Concrete by Frederick M. Goodman.

Topics

The goal of this course is to introduce important topics in the study of Algebra. We will devote most of the course to the study of groups. We will also introduce rings and fields in this course.

Grading

Weekly Homework: 25%

Midterm Exams (Sep 28, Nov 2): 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.

There will be opportunities throughout the semester to earn up to
3 dropped homework grades.

Schedule

The two midterm exams are scheduled for Wednesday, September 28 and Wednesday, November 2. 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**(Aug. 22, 24, 26): Sections 1.1-1.4 on symmetries of polygons.**HW1**Due Wed. Aug. 31: 1.3.1, 1.3.3, 1.4.2, B.4, E.1.*(The notation in the appendices may be different from what you are used to, please let me know if you'd like some of it explained.)*Solutions.

- Aug. 22 Game: Breakout.
*(For reference, the solutions were 253, 764, 861.)* **Week 2**(Aug. 29, 31, Sep 2): Sections 1.5-1.6 on permutations and properties of the integers.**HW2**Due Wed. Sep. 7: 1.5.3 a,b,g,h, 1.5.6, read example 1.5.2 and do exercise 1.5.11, 1.6.3. You can do the calculations in 1.5.3 and 1.5.11 without providing a complete proof. Solutions.

- Aug. 29 Game: Battleship.
- For Friday's class, please read section 1.6. We will go through the highlights together, but you are responsible for the full section.
**Week 3**(Sep. 7, 9): Sections 1.6 - 1.7 on prime numbers and modular arithmetic.

**HW3**Due Wed. Sep. 14: 1.6.9, 1.7.4, 1.7.11, 1.7.14. Solutions.

**Week 4**(Sep. 12, 14, 16): Sections 1.7, 1.9-1.12 on modular arithmetic, counting, groups, and rings.**HW4**Due Wed. Sep. 21: 1.9.1, 1.10.2, 1.10.4, 1.10.9, 1.11.2. Solutions.

- Sep. 16 Activity: RSA Encryption. Solutions.
**Week 5**(Sep. 19, 21, 23): Sections 2.1 and 2.2 on groups and subgroups.

**HW5**Due Wed. Sep. 28: 2.1.5, 2.1.15, 2.2.2, 2.2.7, 2.2.24. Solutions.

- Sep. 23 Activity: Small Groups.
**MT1**Study Guide.

**Week 6**(Sep. 26, 28, 30): Midterm 1, Section 2.3 on the dihedral groups.

- Exam 1, Solutions.

- Sep. 30 Activity: Subgroups and the Dihedral Group.
**Week 7**(Oct. 3, 5, 7): Sections 2.4 and 2.5 on Group Homomorphisms and Cosets.**HW6**Due Wed. Oct 12: 2.3.6, 2.4.4, 2.4.8, 2.4.13, 2.5.4. Solutions.

**Week 8**(Oct. 10, 12, 14): Sections 2.5, 2.6, and 2.7 on Lagrange's Theorem, equivalence relations, and quotient groups.

**HW7**Due Wed. Oct 19: 2.5.7, 2.5.13, 2.6.1, 2.7.9. Solutions.

- Oct. 10: Vocabulary Quiz, Solutions.
- Oct. 12: Activity: Equivalence Relations.
**Week 9**(Oct. 17, 19, 21): Sections 2.7 and 3.1 on isomorphism theorems and direct products.**HW8**Due Wed. Oct 26: 2.7.4, 2.7.6, 2.7.7, 2.7.11. Solutions.

**Week 10**(Oct. 24, 26, 28): Sections 3.1, 3.5, 3.6 on direct product, the Chinese Remainder Theorem, and the Fundamental Theorem of Finitely Generated Abelian Groups.**HW9**Due Wed. Nov 2: 3.1.14, 3.1.15, 3.5.3, 3.6.10. Solutions.

**MT2**Study Guide. Pratice Exam Page (meant to take approximately 20 minutes), Solutions.

**Week 11**(Oct. 31, Nov 2, 4): Midterm 2, Section 5.1 on group actions.

**Week 12**(Nov 7, 9, 11): Sections 5.1-5.3 on group actions.**HW10**Due Wed. Nov 16: 5.1.4, 5.1.6, 5.2.2, 5.3.7. Solutions.

- Nov. 7 Activity: Orbit-Stabilizer Theorem.
**Week 13**(Nov 14, 16, 18): Sections 5.4 and 6.1 on the Sylow theorems and an introduction to ring theory.**HW11**Due Wed. Nov 30: 5.4.1, 5.4.3, 5.4.11, 6.1.5, 6.1.6. Solutions.

- Nov. 18 Activity: Rings and Subrings.
**Week 14**(Nov 28, 30, Dec 2): Sections 6.2-6.4 on ideals, quotient rings, and integral domains.**HW12**Due Wed. Dec 7: 6.2.17, 6.3.7, 6.3.8, 6.4.14. Solutions.

**Week 15**(Dec 5,7): Review.- Dec. 5 Activity: "Pub" Quiz. (Solutions are on second page in blue.)
- Information about the final exam.
- Practice pages
for the final exam. Solutions.

Syllabus

Piazza - This is our course page on Piazza. It is visible only to me and to other students in this class. You can ask and answer course related questions here.

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).