### Algebraic Structures, Spring 2021

Group ring

© 2021 Laure Bukh

Used with permission

#### When:

Mondays, Wednesdays, Fridays 4:30#### Where:

Online via Zoom#### What:

The aim of this course is to introduce algebraic structures that pervade mathematics: groups and rings. We will learn what they are, will see many examples, learn how to reason about them. Topics to be covered include permutation groups, abelian groups, cyclic groups, homomorphisms, quotient groups, group actions, group classification, rings, ring homomorphisms, ideals, integral domains, quotient rings, unique factorization domains, principal ideal domains, and fields.

The prerequisites are being comfortable with reading and writing proofs, and a little of bit of linear algebra.

#### Resources:

- Lecture notes by Samir Siksek. They are well-written, but contain slightly less than what we are going to cover.
*Abstract Algebra*by Dummit and Foote. This is an excellent book for self-study, both at the beginner and more advanced levels.

#### Class format:

This is a remote class. The meeting ID is 932 5786 1340, which is accessible only through CMU Zoom accounts. The password was e-mailed to the registered students on 31st of January. If you add the class after this date or want to audit the class, e-mail me.

To maintain interactive and informal spirit, the students are required to keep their cameras on.

#### Office hours:

The office hours will be at 11:30am on Tuesdays and 8:30am on Thursdays, online via Zoom meeting ID 998 3409 8577 using the same password as for the lectures. The hours are subject to change. I am also available by appointment.

#### Course activities:

There will be weekly homeworks, two mid-terms and and a final. The mid-terms will take place on **March 5th** and **April 9th**. The final exam will be scheduled by the registrar.

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-terms will count for 22% each, whereas the final will count for 41%.

#### Homeworks:

Practice is an integral to learning mathematics. You are encouraged to do as much homework as possible on your own; this way you will learn more. Though collaboration is allowed, you must write the solutions yourself. Turning in solutions that you do not understand will be treated as cheating.

Homework must be neat. Each word must be readable. Anything that you do not want to be graded must be *completely* crossed out.
If in doubt, either re-write solution from scratch or typeset it in LaTeX. Any solution that fails to be neat will receive 0.

In order to ensure academic integrity, each week I will call on some subset of students to explain their solutions to me outside class.

The homework must be submitted via Gradescope.

#### Exams:

Exams are open book. You may use your notes, references, and any inanimate online resources.

You may not use assistance from other people, either in-person or online. This includes e-mail, SMS, any other kind messaging, or posting on online forums/disscussion boards or similar websites.

You must take exam at any time on the exam day, between 12:01am and 11:59pm Eastern time. You may not discuss the exam with anyone (except me) until the exam day ends. This includes even comments such as ``The exam was easy/hard.''.

All exams are self-proctored. You must record yourself using Zoom (audio, video and screen), for the entire duration of the exam.

- Start a new Zoom meeting. Share your screen (entire screen, not just a window). Record it to the cloud.
- Be sure that the recording also shows you, either picture in picture, or separately. There are many settings in Zoom that change this, so please test it before the exam.
- Your recording should start with you identifying yourself, and show you downloading the exam.
- The screen recording should capture the entire screen, and can’t have any time gaps; it should end with you uploading your exam on Gradescope.
- You should stay within your webcam's field of view for the entire exam.
- If any other electronic device is used (tablets / phones), you must also take a screen recording of that device via Zoom for the entire duration of the exam. (The only exception to this is if you only use a device to scan and submit your exam, and the device is
*only*used at the very end where it is clear you are only submitting your exam, then I do not require a screen recording of this device.) - Your recording must remain accessible to me the Zoom server for at least 6 months.
- Additionally, you must download a copy of the recording, and submit it as instructed before the exam. Exams turned in without an accompanying recording will receive no credit.

#### Academic integrity:

Violation of academic integrity include, but are not limited to,

- Not writing solutions independently.
- Turning in solutions that you do not understand.
- Receiving or providing assistance during an exam.
- Discussing the exam with anyone on the exam day.

#### 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 an advanced mathematics 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:

- February 1: Introduction. Groups. Examples. Commutativity. Abelian groups.
*D*. Symmetry._{4}*GL*(_{2}**R**). Homework #1 - February 3:
*GL*(_{2}**R**) as ``symmetries'' of the plane. Elements of*D*as linear transformation. Symmetric groups. Cycles. Products of cycles. Non-commuting cycles. Decomposition of a finite permutation as a product of disjoint cycles._{4}**Z**/m**Z**. - February 5: Uniqueness of identity. Uniqueness of inverses. Subgroups. Examples. Subgroup criterion.
- February 8: Cyclic groups. Cyclic subgroups. Order of a group. Order of an element. Homomorphisms. Isomorphisms. Vector spaces are groups too. Homework #2
- February 10: Examples of isomorphisms. Automorphisms. Some self-proving theorems. Group actions.
- February 12: Actions of a group on itself. Restriction of actions. Conjugation. Orbits. Cosets. Lagrange's theorem.
- February 15: Index. Euclidean algorithm. Group (
**Z**/*n***Z**)^{*}. Fermat's little theorem. Euler's theorem. Product of groups. Homework #3 - February 17: Kernels. Normal subgroups. Quotients. Simple example. Quotients are well-defined.
- February 19: Examples of quotients. Some pictures. Projection map. First isomorphism theorem.
- February 22: Third isomorphism theorem. Computer failure. Homework #4
- February 24: Third isomorphism theorem (again). Simple groups. Groups actions are homomorphisms. Cayley's theorem.
- February 26: Sign of a permutation. 15 puzzle.
- March 1: Alternating group. Conjugation in a symmetric group. Alternating group is generated by
*3*-cycles. Alternating groups are simple. Homework #5 - March 3: Stabilizers. Sizes of orbits. Normalizers. Center. Class equation. Groups of order
*p*.^{2} - March 5:
**Midterm #1** - March 8: Groups of order
*p*(part II). Statement of Sylow's theorems. Product of subgroups. Homework #6^{2} - March 10: Cauchy's theorem for abelian groups. Pre-image of subgroups. Proof of Sylow's theorem (part I).
- March 12: Proof of Sylow's theorem (part II).
- March 15: Applications of Sylow's theorem: groups of order
*pq*and groups of order*12*. About direct products. Homework #7 - March 17: Semidirect products. Group of rigid motions of the plane. Groups of order
*pq*. - March 22: Rings. Examples. Commutative rings. Group of units. Fields. Matrices over arbitrary ring. Quadratic extension of
**Q**. Homework #8 - March 24: Zero divisors. Integral domains. Polynomial rings (informal and formal definitions). Rings of powers series. Group rings.
- March 26: Group rings (part II). Polynomial rings in several variables. Ring homomorphisms. Ideals. Quotient rings.
- March 29: Evaluation homomorphisms. Polynomials and polynomial functions. First isomorphism theorem for rings. Ideals in
**Z**. Maximal ideals. Homework #9 - March 31: Ideals in
**Z**(cont'd). Examples. Fields of prime size. Prime ideals. Complex numbers as a quotient. Euclidean domains. - April 2: Ring extensions. Norm in complex numbers. PID. Euclidean domains are PID. GCD.
- April 5:
*No class*Homework #10 - April 7: Gaussian integers are PID. Other quadratic extensions of
**Z**. Prime and irreducible elements. UFDs. Ascending chain condition. - April 12: UFDs are PIDs. Factorization in Gaussian integers (part I). Homework #11
- April 14: Factorization in Gaussian integers (part II). Sums of two squares in
**Z**. Fields of fractions. - April 19: Content of polynomials. Gauss's lemma. Homework #12
- April 21: Polynomial rings over UFD are UFD.
- April 23: Irreducibility criteria. Chinese remainder theorem (part I).
- April 26: Chinese remainder theorem (part II). Subfields. Field extensions. Degree. Homework #13
- April 28: Prime fields. Adding a root of a polynomial to a field.
- April 30: Algebraic elements. Degree of extending by an algebraic element.
- May 3: Degree of iterated extensions. Q-bar. Examples.
- May 5: More examples. Straightedge and compass constructions (part I).