Instructors Info

Lecture: Maria Angelica Cueto
Email: cueto.5@osu.edu

Recitation: Sjuvon Chung
Email: chung.809@osu.edu

Office Hours (on Zoom)

TBD,
or by appointment.

Time and Location

Live Lecture: M-W-F 11:30am-12:25pm
on Zoom.

Live Recitation: T-R 11:30am-12:25pm
on Zoom.

This is part one of a year-long graduate course on Abstract Agebra. It covers the basic concepts of algebra that are that are necessary for graduate level core mathematics. It is now part of the qualifying exam system and can be used to cover the PhD Breadth Requirement for Algebra. To receive credit for the course as part of the qualifying exam structure, you must obtain an A- or higher grade.

The course will be divided in three parts. In the first part, we will study Groups and their algebraic structures. In the second part, we will focus on Ring Theory, with emphasis on the commutative setting. Finally, in the last part, we will discuss various topics of linear and multilinear algebra, including tensor and exterior algebras. Time permitted, additional topics will be covered.

Content:
  • Group Theory: Monoids, subgroups, cyclic groups, homomorphisms, normal subgroups, factor groups, direct products, group actions, counting lemmas, Sylow theorems, composition series, Jordan-Hölder theorem, solvable and nilpotent groups, semi-direct products, permutation group, simple grups, simplicity of alternating groups.
  • Ring Theory: basic definitions of rings, ideals and modules, ring homomorphisms, commutative rings, integral domains, Chinese remainder theorem, prime and maximal ideals, prime avoidance, rings and modules of fractions, localization, Noetherian and Artinian rings, Hilbert basis theorem, Primary decomposition, PID, UFD, polynomial rings over UFDs, irreducibility criteria, symmetric polynomials, discriminant, modules over PIDs
  • Linear and Multilinear Algebra: Vector spaces, basic operations on vector spaces (direct sum, tensor product, dual vector spaces, Hom-spaces), bilinear and multilinear forms, quadratic forms, positive definite and semidefinite forms, tensor algebra, symmetric and exterior algebra, definition of determinant and minors of an endomorphism, automorphisms preserving a bilinear form (symplectic and orthogonal matrices), polar, Gauss and Jordan decompositions of a matrix.

Prerequisites: Math 5112, or Grad standing.

Syllabus and references

 Main resource: The course will be based on in class material and lecture notes (posted after each live Zoom lecture).

Other textbooks and resources: (for ebooks, access requires connection via an OSU proxy, e.g., by using a VPN or being on campus)
  • Basic Algebra I, II by N. Jacobson, Dover.
  • Algebra by S. Lang, 3rd Edition, Springer.
  • Abstract Algebra by D. Dummit and R. Foote, Wiley.
  • Algebra by N. Bourbaki (Chapters 1-3), Springer.
  • Introduction to Commutative Algebra by Atiyah and Macdonald, Addison Wesley.
  • Commutative Algebra with a view toward Algebraic Geometry by David Eisenbud. Available online through OSU library.
  • Expository Notes by Keith Conrad. Available here.

Back to Top

Exams
  • Midterm 1: Friday October 2, 2020 (online). Topics: Group Theory
  • Midterm 2: Monday November 9, 2020 (online). Topics: TBD
  • Final exam (CUMULATIVE): Friday December 11, 2020 (10:00am-11:45am, held online). This date is tentative.

Back to Top

Homework

 There will be 13 homework problem sets. These will be posted as the semester progresses.
Homeworks will be submitted in pdf format via Carmen by the due date. No late homework will be accepted without medical excuse. Assigments must be prepared in LaTeX, using the template provided below.
Participants are encourage to work in teams, but individual solutions must be submitted for grading and credit.

Back to Top

Tentative Schedule

The following is the schedule of topics that we plan to cover each week (it is subject to change). For a list of topics cover each class, see the corresponding handwritten notes in the section entitled Lectures.

WeekTopics
1 Definitions (Groups, homomorphisms, normal subgroups). Quotient groups. Basic Isomorphism Theorems.
2 Presentation of a group. Groups actions. Counting lemmas.
3 Sylow Theorems. Simple groups.
4 Semidirect products. Short exact sequences. Composition series.
5 Jordan-Hölder theorem. Derived groups. Solvable and nilpotent groups. Simplicity of alternating groups.
6 Definitions (rings, ideals). Isomorphism theorems. Characteristic.
Modules - direct sums, products, homomorphisms, exact sequences.
7 Commutative rings. Prime and maximal ideals. Integral domains.
Rings of fractions. Prime avoidance, Chinese Remainder Theorem.
8 Localization, Noetherian rings, Hilbert Basis Theorem. Nilpotent and Jacobson radicals.
9 Artinian rings, Hensel's lemma, Primary decomposition.
10 Free modules, torsion modules, PIDs, modules over PIDs.
11 Modules over PIDS, Smith normal forms, rational normal forms and Jordan normal forms of matrices.
12 Nakayama's Lemma, Cayley-Hamilton theorem. Vector spaces. Basic operations on vector spaces: direct sum, dual vector spaces, Hom-spaces.
13 Bilinear and multilinear forms. Tensor products. Hom-tensor adjointness, canonical tensors.
14 Tensor, Symmetric and Exterior algebras.
Determinants and minors of endomorphisms. Permanents.
15 Bilinar and quadratic forms, symmetric and skew-symmetric forms, Sylvester's theorem. Gaussian decomposition theorems for matrices.

Back to Top

Lectures

Back to Top