Math 5286H: Fundamental Structures of Algebra II (Spring 2020)

Lectures: MWF 10:10-11:00 in Vincent Hall 2.

Instructor: Gregg Musiker (musiker "at"

Office Hours: Mondays 3:30-4:30, Wednesdays 11:00-12:00, Fridays 2:30-3:30 in Vincent 251; also by appointment.

Course Description:

This is the second semester of a course in the basic algebra of groups, rings, fields, and vector spaces. Roughly speaking, the Fall and Spring semesters should divide the topics as follows:

Fall: Vector spaces, linear algebra, group theory, symmetry, and the Spectral Theorem

Spring: Simple Groups, the Sylow Theorems, Rings, fields, and Galois theory. Modules if time permitting

Prerequisites: Some previous exposure to linear algebra (vectors, matrices, determinants) (such as from 2243, 2373, or 2573) would help. Also, one should either have the ability to write and read mathematical proofs (such as from 2283, 2574, or 3283), or have the desire and drive to learn how.

Required text: Algbera, by Michael Artin, (2nd edition 2011, Prentice Hall).

Fall: some (but not all) of Chapters 1-8;

Spring: some (but not all) of Chapters 7, 11-16

Other useful texts (On reserve in math library)
Title Author(s) Year
Introduction to abstract algebra W. Keith Nicholson 2007
Contemporary abstract algebra Joseph A. Gallian 1994
Algebra Thomas Hungerford 1980
Abstract Algebra Dummit and Foote 2004
Abstract algebra: theory and applications Thomas Judson 1994


  • Homework (50%): There will be 5 homework assignments due approximately every other week in class. The due dates are listed below, and the first homework assignment is due on Monday February 3rd.

    I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates their collaborators. Late homework will not be accepted. Early homework is fine, and can be left in my mailbox in the School of Math mailroom in Vincent Hall 107. Homework solutions should be well-explained; the grader is told not to give credit for an unsupported answer. Complaints about the grading should be brought to me.

  • Exams (15% each): There will be 2 take-home exams, handed out on February 17th (due February 24st) and March 27th (due April 3rd). Each will be open book, open notes, and with calculators allowed. However, for these exams, you are not allowed to consult other electronic sources, such as the internet, and you are not allowed to collaborate or consult with other students or other human sources. These exams are to be collected in class.

  • Final Exam (20%): The final exam will be take-home as well, under the same policies as above, handed out by April 27th, to be turned in during class on May 4th.

  • Class Participation:

    Participation in class is encouraged. Please feel free to stop me and ask questions during lecture. Otherwise, I might stop and ask you questions instead.

    Tentative Lecture Schedule

  • (Jan 22) Lecture 1: Welcome Back and Definition of a Ring and Polynomial Rings (Artin 11.1-11.2)
  • (Jan 24) Lecture 2: More on Polynomial Rings and Ring Homomorphisms (Artin 11.2-11.3)
  • (Jan 27) Lecture 3: The Substitution Principle and Ideals (Artin 11.3)
  • (Jan 29) Lecture 4: Quotient Rings (Artin 11.4)
  • (Jan 31) Lecture 5: Adjoining Elements (Artin 11.5)
  • (Feb 3) Lecture 6: Integral Domains and Product Rings (Artin 11.6-11.7)
  • (Feb 5) Lecture 7: Idempotents and Fractions (Artin 11.6-11.7)
  • (Feb 7) Lecture 8: Maximal Ideals and Algebraic Geometry (Artin 11.8-11.9)
  • (Feb 10) Lecture 9: Factoring Integers and Unique Factorization Domains (Artin 12.1-12.2)
  • (Feb 12) Lecture 10: Euclidean Domains and Gaussian Integers (Artin 12.2)
  • (Feb 14) Lecture 11: Gauss Primes and Factoring Polynomials (Artin 12.2, 12.5)
  • (Feb 17) Lecture 12: Gauss' Lemma and Factoring Integer Polynomials (Artin 12.3-12.4)
  • (Feb 19) Lecture 13: Quadratic Number Fields: Algebraic Integers and their Factorizations (Artin 13.1-13.2)

  • Homework assignments and exams
    Assignment or Exam Due date Problems from Artin text,
    unless otherwise specified
    Homework 1 Monday 2/3 Chapter 11: # 1.1, 1.3, 1.6 (a), 1.8, 1.9, 3.2, 3.3, 3.8, 3.9, 3.12, 3.13, 4.1, 4.2, 4.3, 5.1, 5.4
    Homework 2 Monday 2/17 Chapter 11: # 6.4, 6.8, 7.1, 7.2, 7.3, 7.5, 8.2, 8.3, 8.4
    Chapter 12: # 2.1, 2.6, 2.7, 2.8, 4.7, 5.1, 5.3
    Exam 1 Monday 2/24
    Homework 3 Friday 3/6
    Homework 4 Friday 3/27
    Exam 2 Friday 4/3
    Homework 5 Monday 4/20
    Final Exam Monday 5/4

    Class Policies:

  • S/N Grade: If you are registered S/N, I will submit a grade of S if your letter grade is C or above, and otherwise a grade of N.

  • Incomplete grade: A grade of 'I' will only be considered when failure to complete all course requirements is for reasons beyond the student's control. The minimum requirement for an incomplete grade is a substantial amount of course work completed at the level of C or better. An 'I' grade also requires a written agreement between the student and the instructor on how the missing work will be completed. After one year, an 'I' turns into an 'F' if the course work is not completed. Any arrangement for an incomplete grade can only be considered before the final exam.

  • Disability Accommodations: Disability services promotes access and equity for everyone at the U of M. If you are registered with DS and would like to discuss accommodations, please contant the instructor as soon as possible. If you require accommodations, but are not registered with DS, please contact their office.

  • To add or drop the course: For the various rules and deadlines for adding or dropping the course, visit One-Stop.

  • Scholastic Misconduct: Academic dishonesty in any work for this course will be grounds for awarding a grade of 'F' for the entire course. University policies regarding academic dishonesty, credit, workload expectations, and grading standards are available here and here.