### Algebra, Fall 2021

Cours d'Arithmétique

#### When:

Mondays, Wednesdays, Fridays 10:10#### Where:

Doherty Hall 1117 [map] [classroom photo]#### What:

Abstraction is a way to strip the irrelevant. Algebra is the language of abstraction.

This course is aimed at students who are already somewhat familiar with abstract algebra, and want to deepen their knowledge. Besides the essentials of groups, ring, and fields, I hope also to give a taste of algebraic geometry.

#### Resources:

There is no required text. The book that most closely matches the course is *Abstract Algebra* by Dummit and Foote. It is an excellent text to learn
from.

#### Class format:

The class is expected to be conducted fully in person. Should the university decide to switch to remote instruction, the students should be ready to use Zoom software.

#### Office hours:

The office hours will be at 2:30pm–3:30pm on Thursdays in Wean 6202. I am also available by appointment.

#### Course activities:

There will be several homeworks, a take-home mid-term (October 15–17) and an in-class final.

Students are expected to fully participate in the class. Discussions during the lectures are encouraged.

Homework will count for 15% of the grade. The mid-term and the final will count for 35% and 50%.

Collaboration on homeworks is allowed, but all writing must be done independently. Collaboration on take-home tests is forbidden. Violators will receive a failing grade for the course, and will be subject to disciplinary actions as explained in the student handbook.

Homework must be submitted via Gradescope.
The homework must be submitted by 10:00am of the day it is due. For late submissions, send them via e-mail to me. For each *minute* past the deadline,
the assignment grade is reduced by 10%.

#### Warning:

This syllabus is more likely to change than a syllabus during a non-pandemic semester. I will strive to minimize disruptions, and will communicate any changes promptly via e-mail (and in class if possible).

#### Staying sane and healthy:

This is a graduate course. It is designed to challenge your brain with new and exciting mathematics, not to wear your body down with sleepless nights. Start the assignments early, and get good nutrition and exercise. Pace yourself, for semester is long. If you find yourself falling behind or constantly tired, talk to me.

#### Lectures:

- August 30: Class overview. Homomorphisms. Group actions. Orbits. Homework #1
- September 1: Cosets as orbits. Stabilizers. Orbit-stabilizer theorem.
*p*-groups. Sylow I (beginning). - September 3: Sylow I (finish). Sylow II. Normal subgroups. Quotients. Simple groups. Classification of finite simple groups (gist).
- September 8: Product of subgroups. Sylow III. Groups of order
*pq*. Semidirect products (part I). Homework #2 - September 10: Semidirect products (part II). Dihedral groups. Rigid motions. Holomorphms. Free group. Universal property of free groups.
- September 13: Relations. Presentations. Commutator subgroup. Abelianization. Direct products. Direct sums. Normal series. Derived series. Solvable groups. Homework #3
- September 15: Rings. Zero divisors. Units. Integral domains. Rings of matrices. Polynomial rings. Power series. Laurent series. Group rings. Ring extensions.
- September 17: Ring homomorphisms. Ideals. Quotients. Ideals generated by subsets. Principal ideals. Euclidean domains.
- September 20: Gaussian integers are an Euclidean domain. PID. Euclidean domains are PID. Irreducibles. Primes. Maximal ideals. Homework #4