Algebra, Fall 2021

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 (December 6).

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
• September 22: UFDs. PIDs are UFDs. Localization.
• September 24: GCD. Content of polynomials. Gauss's lemma. UFDs over polynomial rings are UFD. Factorization in Gaussian integers (part I).
• September 27: Factorization in Gaussian integers (part II). Sums of two squares. Number of roots of a polynomial over a field. Systems of polynomials equations and polynomial ideals. Leading coefficients. Hilbert's basis theorem (part I). Homework #5
• September 29: Hilbert's basis theorem (part II). Monomial orderings. Division algorithm in several variables.
• October 1: Monomial ideals. Gröbner basis. Syzygies. Buchberger's criterion (part I).
• October 4: Buchberger's criterion (part II). Computing Gröbner basis. Elimination ideals. Modules. Homework #6
• October 6: Examples of modules. Homorphisms and quotients of modules. Submodules, direct sums, free modules. Tensor product (construction and the universal property).
• October 8: Examples of tensor products. Tensor products and direct sums. Associativity of tensor products. Multilinear maps. Alternating maps.
• October 11: Tensor algebra. Exterior powers. Exterior algebra. Homework #7
• October 13: Exterior algebra of finite-dimensional vector spaces. Plücker embedding. Torsion elements. Torsion-free modules.
• October 15: Structure theorem for modules over PID (part I).
• October 18: Structure theorem for modules over PID (part II).
• October 20: Structure theorem for modules over PID (part III). Homework #8
• October 22: Finite subgroups of multiplicative group of a field. Cayley–Hamilton theorem. Jordan canonical form. Prime fields. Characteristic. Field extensions. Adjoining a root.
• October 25: Degree of an extensions. Algebraic elements. Homework #9
• October 27: Straightedge and compass construction. Splitting fields.
• October 29: Uniqueness of splitting fields. Algebraically closed fields. Axiom of choice and Zorn's lemma. Existence of maximal ideals. Construction of algebraically closed fields. Notes on the axiom of choice
• November 1: Algebraic closure. Construction of an algebraic closure. Separable polynomials and separable extensions. Formal derivatives. Example of a non-separable extension. Homework #10
• November 3: Finite fields. Cyclotomic polynomials. Cyclotomic extensions. Fixed fields.
• November 8: Action of automorphism group on roots. Galois extensions. Linear characters. Independence of characters. Homework #11
• November 10: Degree of a fixed field. Normality. Galois correspondence.
• November 12: Normal extensions and normal subgroups. Examples. More about finite fields. Simple extensions.
• November 15: More about simple extensions. Galois groups of compositum. Homework #12
• November 17: Simple root extension. Cyclic extension. Solvability in radicals (part I).
• November 19: Solvability in radicals (part II). Bring radical. Varieties.
• November 22: Hilbert's Nullstellensatz. Weak Nullstellensatz. Sylvester's matrix. Resultant.
• November 29: Properties of the resultant. An extension lemma.
• December 1: Proof of weak Nullstellensatz. Bezout's theorem (statement sketch). Bezout's theorem in the plane (inequality form, part I).
• December 3: Bezout's theorem in the plane (inequality form, part II). A few words on intersection multiplicity.