Faculty of Science Course Syllabus
Department of Mathematics & Statistics

MATH 4055/5055: Advanced Algebra II (Winter 2022)

Instructor: Theo Johnson-Freyd, theojf@dal.ca.
Lectures: Mondays, Wednesdays, and Fridays, 11:35-12:25 Atlantic Time. LSC-Common Area C212 or https://pitp.zoom.us/j/96394057508.
Office Hours: Tuesdays and Wednesdays 13:05-14:25 Atlantic Time, or by appointment. Chase 214 or https://pitp.zoom.us/j/95588046787.
Course website: http://categorified.net/22Winter5055/.

This course is governed by the academic rules and regulations set forth in the University Calendar and by the Senate.

Course Description

This course covers field theory, field extensions, Galois theory and applications. An introduction to representation theory of finite groups will be included as time permits.

Prerequisites

MATH 4045/5045: Advanced Algebra I

Course materials

The main textbook for the class is Algebra by T.W. Hungerford; we will cover chapters V and VI. Supplementary material will be drawn from Abstract Algebra by D. Dummit and R. Foote (chapters 13, 14, 18, and 19). It is recommended but not required that students own copies of both books; they are available from the Dalhousie Bookstore and many online vendors.

Homework, notes, and so on will be available on the Course Content page.

Course Delivery

Lectures will be given over Zoom until January 31. The Zoom link will be https://pitp.zoom.us/j/96394057508, and the password is "Dal-1234" except with the numbers replaced by the course number. You must be logged into your Zoom account to join the meeting. If for some reason you cannot access Zoom, contact the instructor as soon as possible.

Office hours are Tuesdays and Wednesdays 13:05-14:25 Atlantic Time, or by appointment, and will also be held over Zoom until at least January 31. The Zoom link will be https://pitp.zoom.us/j/95588046787, and the password is "Chase-123" except with the numbers replaced by the instructor's office number.

We will transition to in-person meetings beginning January 31. If/when we do transition to in-person meetings, the class will meet in LSC-Common Area C212, and office hours will be in Chase 214.

This is a very small class. Although attendance is not mandatory, your presence or absence will certainly be noticed by all participants. Lectures will not be recorded. When possible, please keep your cameras turned on during the lectures in order to maintain an interactive online environment.

Course Assessment

Homework assignments be distributed roughly weekly. You should complete every question, and rewrites will be accepted. I encourage you to work together to complete your homework assignments. Studies have shown that social ties are a main predictor of success in STEM classes. Although you are encouraged to work together, the homework you submit must be written by you individually. There are many online resources offering solutions to homework at all levels. If you choose to use such resources, please be cautious: they often provide too detailed an answer, and students can trick themselves into thinking that they understand more than they do by copying those answers.

The final exam for the course will be almost identical to the Algebra Comprehensive Exam Part II. (The Comprehensive will include one or two additional questions on material from Math 3031.) There will not be a midterm.

Each student is required to give a presentation (roughly 20 minutes) on a topic related to the course, to be selected in consultation with the instructor. Some topics people have spoken on in the past: Burnside's Lemma (a link between algebra and combinatorics), What do free groups and group rings have in common? (the use of category theory in algebra), and The non-existence of a construction for trisecting an angle (a link between algebra and geometry). Presentations will be given throughout the course.

Additionally, each student should prepare a short (roughly 5 single-spaced pages) expository article surveying the same material as their presentation. The final draft of the article is due at the end of the semester, and it will be evaluated both on content and writing style. There will be an opportunity to submit drafts for feedback a few weeks before the end of the semester.

Component weighting

Homework: 30%.
Presentation: 20%.
Expository article: 20%.
Final exam: 30%.

Conversion of numerical grades to Final Letter Grades follows the Dalhousie Common Grade Scale

A+ (90-100)
A (85-89)
A- (80-84)
B+ (77-79)
B (73-76)
B- (70-72)
C+ (65-69)
C (60-64)
C- (55-59)
D (50-54)
F (<50)