Math 361, Spring 2020

From cartan.math.umb.edu

Course information[edit]

  • See the syllabus for general information and the schedule of readings.
  • Class meets Mondays, Wednesdays, and Fridays, 11:00 a.m.-11:50 a.m., in W-1-31.
  • Textbook: John Fraleigh, A First Course in Abstract Algebra, Seventh Edition.
  • Instructor: Steven Jackson.
  • Office: W-3-154-27
  • Office hours: by appointment.
  • E-mail: Steven.Jackson@umb.edu.
  • Telephone: (617) 287-6469.
  • As of March 13, this course has shifted to online delivery via this page. Please see below for details.

Important dates[edit]

  • Weekly quizzes happen on Wednesdays during the last ten minutes of class. The first quiz is on Wednesday, February 5.
  • First midterm: Monday, March 2.
  • Second midterm: Monday, April 13.
  • Final exam: Due at 5:00 p.m. on Monday, May 18 (instructions will be e-mailed on Wednesday, May 13).

Online instruction[edit]

Due to the novel coronavirus pandemic, the University has mandated that all courses move to online instruction for the remainder of the Spring semester.

I will be recording video lectures and posting them on this page. Please watch them in a timely manner, noting any questions and e-mailing them to me so that I can address them at the start of the next lecture. You can find all the lectures at the bottom of this page.

We will continue with weekly homework assignments as usual. However, we will have no more quizzes. Please do not wait until the day before the exam to watch the lectures and do the homework. The pandemic is placing many unusual responsibilities on many people; avoiding procrastination without the prompt of the weekly quiz is now one of your unusual responsibilities.

The dates of the remaining exams remain unchanged. However, the precise mode by which the exams will be administered has yet to be determined.

How to use this page[edit]

Below you will find links to the weekly assignment pages. Each of these pages is editable by anyone in the class, so apart from telling you what problems to work on they are excellent spaces in which to ask questions. (If you are very shy you may ask your questions privately, either by email or in person. But we will all work more efficiently if you ask them on the wiki, so that each question only needs to be answered once.) It is also extremely helpful to try to answer questions posed by other students. I will monitor these pages to ensure that no wrong answers go uncorrected.

If you are not already familiar with them, you may wish to read about wiki markup and typesetting mathematics. Also, you may wish to add this page and the assignment pages to your watchlist using the link in the upper right corner of each page, then change your preferences to enable e-mail notifications; this way you will know about page activity without constantly re-checking all the pages.

Weekly assignments[edit]

Recorded lectures[edit]

  • Friday, March 20 (to replace the class cancelled on Friday, March 13). Topics: Evaluation morphisms; the universal mapping property of $R[x]$; polynomial long division.
  • Monday, March 23 Topics: Theorem on polynomial long division; remainder of $f$ modulo $g$; calculation of remainders.
  • Wednesday, March 25 Topics: Arithmetic in $F[x]/\left\langle m\right\rangle$.
  • Friday, March 27 Topics: Divisibility relation on $D$; associate relation on $D$; associate classes in $D$.
  • Monday, March 30 Topics: Irreducible elements of an integral domain; irreducible polynomials.
  • Wednesday, April 1 Topics: Prime elements of a domain; relationship between primeness and irreducibility; ideals of $\mathbb{Z}$ and of $F[x]$.
  • Friday, April 3 Topics: Prime ideals and maximal ideals.
  • Monday, April 6 Topics: Irreducibility and primeness in PIDs; unique factorization; divisor chains.
  • Wednesday, April 8 Topics: Example of a divisor chain; theorem characterizing unique factorization domains; Fundamental Theorem of Arithmetic; theorem on unique factorization of polynomials.
  • Friday, April 10 Topics: Practical methods of factorization; Sieve of Eratosthenes; factorization of polynomials over $\mathbb{Z}_p$; construction of finite fields; monic polynomials.
  • Wednesday, April 15 Topics: Field extensions.
  • Friday, April 17 Topics: Kronecker's Theorem; algebraic and transcendental elements; the annihilator and the minimal polynomial.
  • Wednesday, April 22 Topics: Subextensions; subextension generated by a subset; finitely generated extensions; simple extensions; classification of simple extensions.
  • Friday, April 24 Topics: Symbolic computation.
  • Monday, April 27 Topics: Vector spaces; linear combinations and spans; linear relations and linear independence; bases.
  • Wednesday, April 29 Topics: Dimension of a vector space; dimension of $F[x]/\left\langle m\right\rangle$ over $F$; coordinates; homomorphisms and isomorphisms of vector spaces.
  • Friday, May 1 Topics: The Dimension Formula.
  • Monday, May 4 Topics: Algebraic elements and finite-dimensional subextensions; relative algebraic closures; the field of algebraic numbers; algebraic number fields; algebraically closed fields.
  • Wednesday, May 6 Topics: Algebraically closed fields; algebraic closure of a field.
  • Friday, May 8 Topics: Splitting field of a polynomial; practical construction of splitting fields; existence and uniqueness of splitting fields.
  • Monday, May 11 Topics: Finite fields.
  • Wednesday, May 13 Topics: Summary of the course; resources for further study.
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