Instructors Info

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

Recitation: Soumya Sankar
Email: sankar.40@osu.edu

Office Hours

Fridays 4:00-5:15pm
in Cockings Hall CH) 240
or by appointment on Zoom.

Time and Location

Lecture: M-W-F 11:30am-12:25pm
in Derby Hall (DB) 048.

Recitation: T-R 11:30am-12:25pm
in Derby Hall (DB) 048.

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 class).

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.

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Exams
  • Midterm 1: Friday October 1, 2021 (in class). Topics: Group Theory
  • Midterm 2: Friday November 12, 2021 (in class). Topics: Ring Theory
  • Final exam (CUMULATIVE): Thursday December 16, 2021 (10:00am-11:45am, in DB 048, usual class location)

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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.
You are encourage to prepare you assignements in LaTeX, using the template provided below. Alternatively, you can submit pdf-files of your legible handwritten notes in two ways which will not cause major disruptions. For those of you who have access to a tablet, this should be simple (e.g. by using Notability on an Ipad). Otherwise, you can scan your handwritten solutions with your smartphone and generate a pdf. There are several apps and tutorials online for this (see, e.g. this one).
Participants can work in teams, but individual solutions must be submitted for grading and credit. The best homework solutions will be posted (with author's permission).

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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
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.
14 Canonical tensors. Tensor, Symmetric and Exterior algebras.
15
Universal properties. Determinants and minors of endomorphisms. Permanents. Gaussian decomposition theorems for matrices. Bilinear forms, symmetric and skew-symmetric forms.
16 Sylvester's theorem. Classification of bilinear real symmetric and skew-symmetric forms.
Automorphisms preserving bilinear forms: symplectic and orthogonal matrices. Polar decomposition.

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Lectures
  • Lecture 1 (Introduction, course overview and examples), August 25, 2021.
  • Lecture 2 (Subgroups, Normal subgroups, group structures on space of maps; aside: the quaternion group Q8), August 26, 2021.
  • Lecture 3 (Quotient and cyclic groups; first counting lemma), August 28, 2021.
  • Lecture 4 (Three isomorphism theorems for groups), August 30, 2021.
  • Lecture 5 (Order, exponent, Free groups, universal properties, presentation of groups by generators and relations, examples), September 1, 2021.
  • Lecture 6 (Group actions: definition, orbit, stabilizers, fixed point sets, examples; first counting lemmas), September 3, 2021.
  • Lecture 7 (Group actions II: Counting lemmas, Burnside's Lemma), September 8, 2021.
  • Lecture 8 (Applications of Burnside's lemma and Sylow Theorems), September 10, 2021.
  • Lecture 9 (Sylow Theorems II: examples and applications to classifications of groups), September 13, 2021.
  • Lecture 10 (Semidirect Products I), September 15, 2021.
  • Lecture 11 (Semidirect Products II; exact sequences; sections and retractions for short exact sequences), September 17, 2021.
  • Lecture 12 (Short exact sequences; Direct and Semidirect Products via trivial and split short exact sequences; Composition Series), September 20, 2021.
  • Lecture 13 (Schrier's Theorem and Zassenhaus' Lemma; Jordan-Hölder series), September 22, 2021.
  • Lecture 14 (Derived series and solvable groups; Lower central series and nilpotent groups; examples; characterizations of solvable and nilpotent groups via composition series), September 24, 2021.
  • Lecture 15 (Characterization of solvable and nilpotent groups; subgroups and quotient groups of solvable/nilpotent groups; structure theorem for finite nilpotent groups), September 27, 2021.
  • Lecture 16 (Basics on Rings I: definitions, ideals, special rings, examples), September 29, 2021.
  • Lecture 17 (Basics on Rings II: Homomorphisms, Isomorphism theorems; Algebra of ideals; characteristic of a ring; ), October 4, 2021.
  • Lecture 18 (Modules: definitions, homomorphisms, direct product, direct sum, short exact sequences), October 6, 2021.
  • Lecture 19 (Chinese Remainder theorem; prime and maximal ideals, existence of maximal ideals), October 8, 2021.
  • Lecture 20 (Maximal ideals are prime ideals, prime avoidance; local rings, examples of local rings), October 11, 2021.
  • Lecture 21 (Characterization of local rings, multiplicatively closed sets, ring of fractions), October 13, 2021.
  • Lecture 22 (Ideals on rings of fractions, examples, Localization), October 18, 2021.
  • Lecture 23 (Localizations of domains, Modules of fractions; exactness; Noetherian rings: first properties), October 20, 2021.
  • Lecture 24 (Noetherian rings and Noetherian modules; examples; basic properties), October 22, 2021.
  • Lecture 25 (Hilbert's Basis Theorem; Artinian rings: examples, first properties: dimension 0 and finitely many maximal ideals), October 25, 2021.
  • Lecture 26 (Artinian rings II: Structure theorem; Artinian rings are Noetherian), October 27, 2021.
  • Lecture 27 (Artinian rings and primary decomposition: definitions, examples of primary ideals), October 29, 2021.
  • Lecture 28 (Artinian rings and primary decomposition II: irreducible ideals, finiteness of minimal primes associated to an ideal in a Noetherian ring; Noetherian, dimension zero rings are Artinian; uniqueness of radicals of primary components for reduced primary decompositions in the Noetherian setting; uniqueness features, associate vs. minimal primes, uniqueness of minimal primary components), November 1, 2021.
  • Lecture 29 (Primary decomposition for PIDs; PIDs are UFDs), November 3, 2021.
  • Lecture 30 (Modules over PIDs I: Free modules over commutative rings, ranks, submodules of free modules; torsion for modules), November 5, 2021.
  • Lecture 31 (Modules over PIDs II: Splitting of f.g. modules over PIDs, Decomposition of torsion modules over PIDs into p-torsion components; Classification theorems), November 8 and 9, 2021.
  • Lecture 32 (Moduldes over PIDs III: Rational normal forms of matrices), November 10, 2021.
  • Lecture 33 (Jordan canonical forms of matrices with complex entries, Cayley-Hamilton Theorem for matrices over a field ), November 15, 2021.
  • Lecture 34 (Consequences of Cayley-Hamilton, Nakayama's Lemma; Basics on Linear algebra: vector spaces, Hom-spaces, bases, direct sums, dual vector spaces), November 17, 2021.
  • Lecture 35 (Double duals and inclusion V* ↪ (V*)*, bilinear maps and tensor products), November 19, 2021.
  • Lecture 36 (Explict construction of tensor products, bases for tensor products, dimension of tensor products, hom-tensor adjointness), November 22, 2021.
  • Lecture 37 (Tensor algebras, symmetric and exterior algebras as quotients, dimensions of symmetric and exterior products), November 29, 2021.
  • Lecture 38 (Symmetric and exterior algebras as subalgebras of the tensor algebra, universal properties, tensor, symmetric and exterior powers of linear maps, compatibility with direct sums), December 1, 2021.
  • Lecture 39 (Determinants and minors; permanents), December 3, 2021.
  • Lecture 40 (Bilinear forms: first properties, degenerate vs. non-degenerate froms; Symmetric, skew-symmetric and alternate bilinear forms; matrix representation of bilinar forms over finite dimensional vector spaces; towards Sylvester's theorem), December 6, 2021.
  • Lecture 41 (Classification of bilinear real symmetric forms; Sylvester's theorem; classification of bilinear symmetric and skew-symmetric forms over finite dimensinal vectors spaces over fields with characteristic not 2), December 8, 2021.

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