Rings and Fields (syllabus)

Aluffi's textbook
Euclidean domain
Richard Dedekind
Prime ideals
Description
As a first undergraduate course in modern algebra, we study rings—an algebraic structure in which addition and multiplication are defined and have properties similar to those operations on the integers. This theory unifies and generalizes many examples from geometry and number theory. This course is intended for all students majoring in mathematics.
Instructor
Gregory G. Smith (512 Jeffery Hall, ggsmith@mast.queensu.ca)
Lectures
(slot 002)
Monday at 09:30–10:20 in 100 Kinesiology
Wednesday at 08:30–09:20 in 100 Kinesiology
Thursday at 10:30–11:20 in 100 Kinesiology
Tutorial
Wednesday at 16:30–17:20 in 201 Jeffery Hall (a.k.a. the Math Help Centre)
 
Assessment
The course grades will be computed as follows:
  • 30% Homework
  • 30% Midterm
  • 40% Exam
Homework
Problem sets are posted in PDF on the lectures webpage. Your browser can be trained to open these files with the free program Acrobat Reader (or other PDF viewer). Solutions to each problem set will be submitted via the Crowdmark system. Instructions for using this software are available. The solution to each problem must be uploaded separately. Solutions are due on Fridays before 17:00; late homework will receive no credit. Your best ten of twelve solution sets will determine your homework grade.
Midterm
There will be a two-hour midterm on Wednesday, 1 March 2023 at 19:00–21:00 in Walter Light Hall AUD (Room 205).
Exam
There will be a three-hour exam on Wednesday, 26 April 2023 at 09:00–12:00 in Bartlett Gym.
 
Learning Outcomes
Upon successful completion of the course, students will be able to do the following:
  • Perform accurate and efficient computations with integers and polynomials involving quotients, remainders, divisibility, greatest common divisors, primality, irreducibility, and factorizations.
  • Define and illustrate basic concepts in ring theory using examples and counterexamples.
  • Describe and demonstrate an understanding of equivalence classes, ideals, quotient rings, ring homomorphisms, and some standard isomorphisms.
  • Recognize and explain a hierarchy of rings that includes commutative rings, unique factorization domains, principal ideal domains, Euclidean domains, and fields.
  • Write rigorous solutions to problems and clear proofs of theorems.
To develop these abilities, students are expected to
  • independently read the course notes or at least one textbook,
  • regularly participate in the lectures and tutorials,
  • persistently complete all homework assignments, and
  • continually discuss mathematics with other students.
Writing
We write to communicate. Please bear this in mind as you complete the homework, the midterm, and the exam. Work must be neat and legible to receive consideration. You must explain your work in order to obtain full credit; an assertion is not an answer.
Academic Integrity
Students are responsible for familiarizing themselves with all of the regulations concerning academic integrity and for ensuring that their assignments and their behaviour conform to the principles of academic integrity. Students are welcome to discuss problems, but should write up the solutions individually. Students must explicitly acknowledge any assistance including books, software, technology, websites, students, friends, professors, references, etc.
Accommodations
The instructor is committed to achieving full accessibility for people with disabilities. Part of this commitment includes arranging academic accommodations for students with disabilities to ensure they have an equitable opportunity to participate in all of their academic activities. If you think that you may need academic accommodations, then you are strongly encouraged to contact the instructor and the Queenʼs Student Accessibility Services (QSAS) as soon as possible.
Licensing
Materials generated by the instructor may not be used for commercial advantage or monetary compensation. Some material is clearly copyrighted and may not be reproduced or retransmitted in any form without express written consent. Other material, licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, may be remix, adapt, or build upon it, as long as appropriate credit is given and the new creation is distributed under the identical terms.
Technology
Students are encouraged to use any available technology on the homework, but these aids (including calculators) will not be allowed during the midterm and exam.
Reference
Paolo Aluffi, Algebra: Notes from the Underground, Cambridge University Press, Cambridge, 2021. ISBN: 978-1-108-95823-3