Introduction to Modern Algebra II | UNL Spring 2023

Introduction to Modern Algebra II

Spring 2023

University of Nebraska — Lincoln

Instructor: Eloísa Grifo (please address me by my first name)


Class notes   If you find any typos at all, however small, please let me know.


Basic course info: This is a sequel to Math 817. We will continue to talk about rings, introduce modules, and discuss topics from field theory including Galois theory and finite fields.

Prerequisites: Math 817.

Course expectations: Attendance is expected, as much as possible, but please do NOT attend class if you are feeling ill or may have been exposed to COVID-19. You are encouraged to work on the problem sets together in groups, or discuss them with me; you should however write up your own solutions. The only other resources you are allowed to use to solve the problem sets are our class notes and the textbook. There will be a midterm and a final exam, both in-person.

Office Hours: You can just drop by my office anytime, or write me an email if you'd like to set up an appointment.
I am definitely going to be in my office and available to talk to you on Mondays 4-5:30+ and Fridays 10-11.
(The + means that I will stay later whenever there are students in my office, but will leave at 5:35ish if I'm alone)

Problem Sets

Instructions: You are welcome to work together on the problems, and I will be happy to give you hints or discuss the problems with you, but you should write up your solutions by yourself. You will submit each problem set on canvas as a pdf file.

Schedule

  • Lecture 1 (Monday, January 23): Why all rings should have a 1
  • Lecture 2 (Wednesday, January 25): Modules: definition and first properties.
  • Lecture 3 (Friday, January 27): Submodules, restriction of scalars, homomorphisms.
  • Lecture 4 (Monday, January 30): Examples and facts about homomorphisms of modules. Hom and End.
  • Lecture 5 (Wednesday, February 1): The Isomorphism theorems for modules.
  • Lecture 6 (Friday, February 3): F[x]-modules.
  • Lecture 7 (Monday, February 6): Module generators and bases.
  • Lecture 8 (Wednesday, February 8): Free modules.
  • Lecture 9 (Friday, February 10): Vector spaces and dimension.
  • Lecture 10 (Monday, February 13): Every vector space has a basis, and every bases for the same vector space has the same number of elements.
  • Lecture 11 (Wednesday, February 15): Every module is a quotient of a free module.
  • Lecture 12 (Friday, February 17): The matrix of a linear transformation. Change of basis.
  • Lecture 13 (Monday, February 20): Elementary operations. Presentations of modules.
  • Lecture 14 (Wednesday, February 22): Presentations of modules.
  • Lecture 15 (Friday, February 24): Finitely generated modules over noetherian rings are finitely presented.
  • Lecture 16 (Monday, February 27): The classification of finitely generated modules over PIDs.
  • Lecture 17 (Wednesday, March 1): Smith Normal form: existence.
  • Lecture 18 (Friday, March 3): Smith Normal form: uniqueness. Proof (stated a few classes earlier) that various matrix operations do not change the isomorphism class of the module presented.
  • Lecture 19 (Monday, March 6): Towards rational Canonical forms.
  • Lecture 20 (Wednesday, March 8): Rational Canonical forms. A short summary of the course so far.

    • Wednesday, March 8, 5 pm: Midterm (Avery 117)

  • Lecture 21 (Friday, March 10): The Cayley-Hamilton Theorem.
  • Lecture 22 (Monday, March 20): A quick review of Rational Canonical Forms. Jordan Canonical Forms.
  • Lecture 23 (Wednesday, March 22): Jordan canonical forms: proof.
  • Lecture 24 (Friday, March 24): Diagonalizability. Field extensions. Notes
  • Lecture 25 (Monday, March 27): Field extensions.
  • Lecture 26 (Wednesday, March 29): Field extensions.
  • Lecture 27 (Friday, March 31): The degree formula.
  • Lecture 28 (Monday, April 3): Algebraic and transcendental elements.
  • Lecture 29 (Wednesday, April 5): Algebraic vs finite extensions.
  • Lecture 30 (Friday, April 7): Algebraic closures. Existence of algebraic closures.
  • Lecture 31 (Monday, April 10): Uniqueness of algebraic closures.
  • Lecture 32 (Wednesday, April 12): Splitting fields: existence and uniqueness.
  • Lecture 33 (Friday, April 13): Splitting fields: examples and degree.
  • Lecture 34 (Monday, April 17): A few more examples of splitting fields. Characteristic. Separability.
  • Lecture 35 (Wednesday, April 19): Separability and perfect fields.
  • Lecture 36 (Friday, April 21): Automorphism groups of field extensions.
  • Lecture 37 (Monday, April 24): More examples of automorphism groups of field extensions. Finite extensions.
  • Lecture 38 (Wednesday, April 26): Galois extensions. Corollaries of Artin's Theorem.
  • Lecture 39 (Friday, April 28): Math 125
  • Lecture 40 (Monday, May 1): The Fundamental Theorem of Galois Theory.
  • Lecture 41 (Wednesday, May 3): Applying the Fundamental Theorem of Galois Theory.
  • Lecture 42 (Friday, May 5): Solvability.
  • Lecture 43 (Monday, May 8): More on polynomials solvable by radicals (or not).
  • Lecture 44 (Wednesday, May 10): The Primitive Elements Theorem. Applications.
  • Lecture 45 (Friday, May 12): Review for the final exam.

    • Wednesday, May 17, 9:30 am: Final Exam (Avery 351)


Helpful resources



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