M380C Algebra: Fall 2020

Day/Time: MWF 12pm-1pm; Location: Zoom & PMA 6.104; Unique: 53985

Instructor
Mirela Ciperiani (mirela at math dot utexas dot edu); Office: PMA 12.164

Office Hours
After each class or by appointment on zoom.

Text
Abstract Algebra, 3rd edition by Dummit and Foote, published by Wiley. We will cover material from parts I-III.
The book should be available at the University Co-op.

Prerequisites
Undergraduate abstract algebra. Contact me for more details if you are not sure whether this course is for you.

Teaching Assistant
TBA(tba at math dot utexas dot edu)
Please contact TBA if you have any questions about the grading of the homework.

Midterm exam
Wednesday, October 21 in class.

Final exam
TBA
All students must take the final at the time scheduled by the university.

Homework
The weekly homework assignments and the grades will be posted on Canvas. The homework will be due on Wednesdays before the beginning of class and sample solutions will be posted on Canvas on the same day.
The lowest homework grade will be dropped.

Grading
Plus/minus grades will be assigned for the final grade in this course.

Homework 30%
Midterm 30%
Final exam 40%

Conflicts
If you have a conflict with any of the exams (for example, due to a religious holiday), you must notify the instructor by the 12-th class day.

Disabilities
Students with disabilities may request appropriate academic accommodations from the Division of Diversity and Community Engagement, Services for Students with Disabilities, 512-471-6259. If you plan on using accommodations, you need to notify the instructor by the 12-th class day.

Syllabus
  1. Groups
    • Definitions and Examples
    • Homomorphisms
    • Subgroups and Quotient Groups
    • Group Actions
    • The Sylow Theorems
  2. Ring Theory
    • Ideals and Homomorphisms
    • Factorization in rings
    • Polynomial rings
  3. Module Theory
    • Basic definitions and Examples
    • Module homomorphisms
    • Tensor products
    • Modules over PIDs
    • Canonical Forms
  4. Possible Supplemental Topics
    • Homological Algebra
    • Basic Commutative Algebra


Recommended reading

This schedule is tentative and will be modified as necessary.

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  Aug. 26, 28   1.1 - 1.5, 2.1- 2.3, 3.1 - 3.2: Groups and subgroups, definition and examples; Cosets and quotient groups
  Aug. 31, Sept. 2, 4   1.6 - 1.7, 3.3, 4.1: Homomorphisms and isomorphisms; Isomorphisms theorems; Group actions: orbits and stabilizers
  Sept. 9, 11   4.2, 4.3, 4.5: Applications of groups actions; Sylow theory
  Sept. 14, 16, 18   4.4, 4.5: Applications of Sylow theory; Representations induced by group actions; Automorphism group
  Sept. 21, 23, 25   4.6, 5.1, 5.5: Automorphisms of Sn; The simplicity of An; Direct and indirect products
  Sept. 28, 30, Oct. 2   5.4, 6.1, 5.2: Solvable and Nilpotent groups; Classification of finite abelian groups
  Oct. 5, 7, 9   7.1 - 7.3: Rings, definition and examples; Ring homomorphisms and quotient rings; Ideals and their properties
  Oct. 12, 14, 16   7.4 - 7.5: Properties of ideals, Rings of fractions
  Oct. 19, 21, 23   7.6, 8.1 - 8.2: The Chinese Remainder theorem, Euclidean Domains, Principal ideal domains
  Oct. 26, 28, 30   9.6, 8.3: Noetherian rings, Hilbert basis theorem, Unique factorization domains
  Nov. 2, 4, 6   9.1 - 9.5: Polynomial rings
  Nov. 9, 11, 13   10.1-10.4: Introduction to modules
  Nov. 16, 18, 20   12.1: Modules over Principal Ideal Domains
  Nov. 23   12.2-12.3: The Rational & Jordan Canonical forms
  Nov. 30, Dec. 2, 4  
  Dec. 7