### Algebraic structures, Spring 2021

Cours d'Arithmétique

#### When:

Mondays, Wednesdays, Fridays 10:40#### Where:

Doherty Hall 2302 [map] [classroom photo]#### Pre-semester questionnaire:

Thanks to everyone who filled the survey. It was very helpful.

#### 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 to give a taste of representation theory and 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:

This is an in-person/remote class. You may choose to attend in-person or via remote technology; you may change the mode of attendance at any time. The connection will be via Zoom. The meeting ID is 948 2762 0577, which is accessible only through CMU Zoom accounts. The password was e-mailed to the registered students on 29th of August. 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 attending remotely will be required to keep their cameras on.

#### Office hours:

The office hours will be at 2:30pm on Thursdays, online via Zoom meeting ID 993 95617 838 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 several homeworks, a take-home mid-term and a take-home final. To help maintain academic integrity, there will be two oral exams for everyone except for the PhD students. The mid-term will be a 4-hour exam at a time of student's choice during October 22-25. The final exam will be a 24-hour exam at a time of student's choice during the finals week.

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% for PhD students. For non-PhD students, the percentages are 25% and 40% respectively. The oral tests for non-PhD students will follow respective take-home tests. They will each be worth 10%. In addition, they will be used to assess that the submitted solutions to take-home tests match students' level of expertise.

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 in LaTeX via e-mail. I want both the LaTeX file and the PDF that is produced from it. The filenames must be of the form `lastname_algebra_homeworknumber.tex`
and `lastname_algebra_homeworknumber.pdf` respectively. Pictures and commutative diagrams do not have to be typeset; a legible photograph of a hand-drawn picture is acceptable.

The homework must be submitted by 10:40am of the day it is due. For each *minute* that it is late, the assignment grade will be reduced by 10%.

#### Auditing the course:

If you want to audit the course, please contact me in advance even if you plan to attend only remotely.

#### 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 31: Class overview. Homomorphisms. Group actions. Orbits.
- September 2: Fixed points. Stabilizers. Class equation.
*p*-groups. First Sylow's theorem. Homework #1 - September 4: Other Sylow's theorems. Application of Sylow's theorems.
- September 7:
*Labor day* - September 9: Groups generated by sets. Free groups (bottom-up definition). Commutators. More commutators. Lower central series. Nilpotent groups.
- September 11: Examples of nilpotent groups. Nilpotent rings.
- September 14: Upper central series. Central series. Finite nilpotent groups.
- September 16: Finite nilpotent groups (finish). Derived series. Solvable groups. Ring basics.
- September 18: Zero divisors. Integral domains. Polynomial rings. Powers series. Infinitesimals. Laurent series. Group rings. Polynomial functions.
- September 21: Primes and irreducibles. Euclidean domains. PIDs. UFDs. Ideal factorization (no proofs). PIDs are UFDs. Homework #2
- September 23: Rings of fractions (aka localization). Universal property. Fields of fractions. Fields of rational functions. Rings of continuous functions on a compact space. Algebra-geometry dictionary.
- September 25: Spaces of polynomials. Localizations of polynomial rings in two variables. Algebraically closed fields. Fundamental theorem of algebra (statement). Gauss's lemma. Polynomial ring over UFD is UFD (part I).
- September 28: Polynomial ring over UFD is UFD (part II). Basic irreducibility criteria. Eisenstein's criterion. Newton polytopes (statements). Noetherian rings. Systems of polynomial equations. Radical.
- September 30: Hilbert's basis theorem. Monomial orderings. The division algorithm for multivariate polynomials.
- October 2: Gröbner bases. Monomial ideals. Syzigies. Buchberger's criterion (part I).
- October 5: Buchberger's criterion (part II). Ascending chain condition. Computing Gröbner bases. Elimination. Homework #3
- October 7: Modules. Examples. Basic constructions. Some universal constructions. Tensor products (part I).
- October 9: Tensor products (part II).
- October 12: Algebras. Quaternions. Tensor algebras. Symmetric and alternating algebras. Antisymmetric and alternating maps.
- October 14: Exterior products and their dimension. Subspaces as elements of the exterior algebra. Modules over PID: submodules of free modules.
- October 19: Torsion elements and torsion modules. Structure theorem for modules over PID (part I). Homework #4. Mid-semester exam rules.
- October 21: Structure theorem for modules over PID (part II).
- October 26: Application of tensor products: zero-error communication.
- October 28: Structure theorem for modules over PID (part III).
- October 30: Chinese remainder theorem. Invariant factors. Rational canonical form.
- November 2: Jordan canonical form. Categories. Products. Coproducts. Field extensions. Characteristic. Homework #5
- November 4: Prime fields. Initial/terminal objects. Opposite categories. Degree of a field extension. Compositum. Algebraic elements and extensions.
- November 6: Degree of of the compositum. Zorn's lemma.
- November 9: Algebraic closure (definition, existence, uniqueness). Number of embeddings for simple extensions.
- November 11: Normal extensions.
- November 13: Separable polynomials. Formal derivatives. Cyclotomic polynomials. Cyclotomic extensions.
- November 16: Finite fields (construction, uniqueness). Field automorphisms. Galois extensions. Galois groups. Fixed field. Homework #6
- November 18: Galois correspondence. Primitive element theorem (infinite field case).
- November 20: Primitive element theorem (finite field case). Normal subgroups of Galois groups. Cyclic extensions.
- November 23: Galois groups of compositum. Radical extension are solvable.
- November 30: A non-solvable quintic. Solvable extensions are almost radical.
- December 2: Varieties. Coordinate rings. Regular functions and maps. Hillbert's Nullstellensatz statement (strong and weak, statements). Resultants (part I).
- December 7: Resultants (part II). Proof of Nullstellensatz.
- December 9: Intersection and unions of varieties. Zariski topology. Projective geometry. Bezout's theorem (statement sketch).
- December 11: Bezout's theorem (inequality form). A few words on intersection multiplicity. Final exam rules.