Algebra, Fall 2022

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–3:20pm on Thursdays and after the class on Fridays in Wean 6202. I am also available by appointment.

Course activities:

There will be several homeworks, a take-home mid-term (October 14–16) and an in-class final (December 15 at 1pm). The students are required to inform me of any conflict for the final exam within one week after the registrar publishes the final exam schedule.

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

Homework will count for 20% of the grade. The mid-term and the final will count for 40% each.

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 andrewid_algebra_homeworknumber.tex and andrewid_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 10:00am of the day it is due. For each minute past the deadline, the assignment grade is reduced by 10%.

Unusually insightful solutions, and other achievements will be appropriately rewarded.

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 29: Class overview. Homomorphisms. Group actions. Homework #1
• August 31: Orbits. Cosets as orbits. Lagrange's theorem. Index. Stabilizers. Orbit-stabilizer theorem.
• September 2: Normal subgroups. Quotients. No converse to Lagrange's theorem. p-groups. Sylow theorems (statement).
• September 5: Labor Day Homework #2
• September 7: Normalizers. Product of subgroups. Sylow theorems (proof).
• September 9: Direct products. Groups of orders p, pq, p²q. Automorphisms. Automorphism group. Semidirect products.
• September 12: Examples of semidirect products. Holomorphms. Rigid motions of Rⁿ. Groups generated by subsets. Groups words. Elementary reductions/expansions. Free group (definition). Homework #3
• September 14: Universal property of the free group. Commutative diagrams. Uniqueness of free groups. Free abelian groups. Normal closure. Presentations. Commutator subgroup.
• September 16: Abelianization. Derived series. Solvable groups. Classification of finite abelian groups (statement). Lower and upper central series. Nilpotent groups. Finite p-groups are nilpotent.
• September 19: Equivalence of lower and upper central series. Rings. Units. Group of units. Zero divisors. Rings of matrices. Polynomial rings. Power series. Laurent series. Group rings. Homework #4
• September 21: Polynomials and polynomial functions. Ring kernels and quotients. Quadratic rings. Generation of ideals, subrings, subfields.
• September 23: Principal ideals. Integral domains. Principal ideal domains (PIDs). Norm in the complex numbers. Euclidean domains. Euclidean domains are PIDs. Gaussian integers are PID.
• September 26: Irreducible elements. Prime ideals, prime elements. Unique factorization domains (UFDs). Noetherian rings. PIDs are UFDs (existence). Homework #5
• September 28: PIDs are UFDs (uniqueness). Conjugation and norms. Factorization in Gaussian integers and sums of two squares (part I).
• September 30: Factorization in Gaussian integers and sums of two squares (part II). Rings of fractions. Localization. Content of polynomials. Gauss's lemma.
• October 3: UFDs over polynomial rings are UFDs. Leading coefficients. Chinese remainder theorem. Lagrange interpolation formula. Hilbert's basis theorem (part I). Homework #6
• October 5: Leading coefficients. Hilbert's basis theorem (part II). Monomial orderings. Division algorithm in several variables.
• October 7: Gröbner bases. Syzygies. Buchberger's criterion. Buchberger's criterion.
• October 10: Computing Gröbner bases. Elimination ideals. Modules. Homework #7 Midterm exam rules.
• October 13: Categories. Opposite categories. Initial and terminal objects. Products and coproducts.
• October 14: Free and torsion-free modules. Rank. Rank of a free module. Submodules of free finitely generated modules over PID.
• October 24: Structure theorem for finitely generated modules over PID (existence and uniqueness). Structure of finite abelian groups. Rational canonical form. Homework #8
• October 26: Jordan canonical form. Tensor products (construction and the universal property). Basis of tensor products. Tensor products and direct sums.
• October 31: More examples of tensor products. Associativity of tensor products. Field extensions. Prime fields. Degree. Finite and algebraic extensions. Adjoining a root. Homework #9
• November 2: Minimal polynomial. Iterated extensions. Compositum. Straightedge and compass constructions (part I).
• November 4: Straightedge and compass constructions (part II). Algebraic closure.
• November 7: Algebraic closure (part II). Homework #10
• November 9: Normal extensions. Separable polynomials. Formal derivatives. Frobenius endomorphism.
• November 11: Finite fields. Separable extensions. Eisenstein criterion. Galois extensions.
• November 14: (Linear) characters. An aside on Fourier transform. Characters are linearly independent. Galois correspondence. Normal extension and normal subgroups. Homework #11
• November 16: Galois group and the roots of a polynomial. Examples. Finite subgroups of multiplicative group of a field. Simple extensions.
• November 18: Cyclotomic polynomials. Möbius function. Irreducibility of cyclotomic polynomials. Cyclotomic extensions.
• November 21: Simple root extensions. Cyclic extensions. Galois closure. Solvability by radicals. Homework #12
• November 28: Insolvability by radicals. Bring radical. Varieties. Hilbert's Nullstellensatz. Weak Nullstellensatz. Resultants (part I).
• November 30: Resultants (part II).
• December 2: Extension lemma. Proof of the Nullstellesatz. Varieties-ideals correspondence.
• December 5: Algebraic curves. Inequality form of Bezout's theorem (part I). Homework #13
• December 7: Inequality form of Bezout's theorem (part II).
• December 9: Projective geometry. Intersection multiplicity.