Math 3GR3 - Abstract Algebra
(Fall 2018)

This course is an introduction to groups and rings with an emphasis on concrete examples.

News Homework SAGE Quizzes Handouts Grading Scheme Schedule Policies

Course Information

Instructor: Adam Van Tuyl

Office: Hamilton Hall 419
Office Hours: M 1:30-2:30, Th 11:30-12:30
Email: vantuyl@math.mcmaster.ca

TA: Graham Keiper

Office: Hamilton Hall 403
Office Hours: Th 2-3PM
Email: keipergt@mcmaster.ca

Place and Time:

Class: Monday and Thursday 3:30-4:20, Tuesday 4:30-5:20 in BSB B136
Tutorial: Friday 3:30-4:20 in KTH B132

Textbook:

• Abstract Algebra: Theory and Applications (2017 Edition)
  by Tom Judson. This is a FREE online textbook. Hard copies can also be found in the bookstore. Note that we will be using the 2017 edition.
• Mathematical Writing (Optional)
  by Franco Vivaldi. This book is optional, and will help you write proofs. An electronic copy is FREE through the McMaster library.

News (Last Updated: Dec. 10, 2018)

The final exam will be on December 17. Here are copies of the review sheets for the midterms and exam.

• Midterm 1 Review Sheet
• Midterm 2 Review Sheet
• Final Review Sheet
• 2017 Final Exam
• 2017 Final Exam Solutions

I will also have office hours during the following times:

• Thursday, Dec. 13, 9-11AM
• Friday, Dec. 14, 9-11AM
• Monday, Dec. 17, 9-11AM

Below is a summary of what we did in class, plus any relevant news and/or information.

• Dec. 4, 2018 Last day of class; I did some review.
• Dec. 3, 2018 I discussed some of the material of Chapter 18 on Euclidean domains and Principal Ideal Domains.
• Nov. 29. 2018 In today's class we looked at Section 17.3 on irreducible polynomials.
• Nov. 27, 2018 Today we proved the division algorithm for polynomials (See Section 17.2).
• Nov. 26, 2018 We started Chapter 17 on polynomial rings. See Section 17.1.
• Nov. 22, 2018 We finished Chapter 16 by going over prime and maximal ideals.
• Nov. 20, 2018 In today's class we proved the first isomorphism theorem for rings.
• Nov. 19, 2018 Midterm 2 has been graded and handed back. Please note that I made the exam out of 28 instead of 30. I also continued our discussion of Section 16.3 by introducing ring homomorphisms.
• Nov. 15, 2018 Midterm 2 was today.
• Nov. 13, 2018 Today I introduced the concept of an ideal, and I gave a number of examples (see Section 16.3).
• Nov. 12, 2018 We continued to look at Chapter 16. We looked at Section 16.2 on integral domains and fields.
• Nov. 8, 2018 We started Chapter 16 on rings. I introduced the main definition, and then gave a number of examples.
• Nov. 6, 2018 We finished looking at Section 11.2 on the Isomorphism Theorems (Section 11.2). Midterm 2 will cover up to today's material.
• Nov. 5, 2018 We looked at the First Isomorphism Theorem for groups today (Section 11.2).
• Nov. 1, 2018 Today we looked at homomorphisms, some of their properties, and I introduced the kernel of a homomorphism (Section 11.1).
• Oct. 30, 2018 We looked at simple groups and some applications of the factor group (Sections 10.1 and 10.2).
• Oct. 29, 2018 I introduced normal subgroups, and I explained how to form the factor group (Section 10.1).
• Oct. 25, 2018 We discussed internal and external direct products (Section 9.2). We then started our discussion of factor groups (Section 10.1).
• Oct. 23, 2018 We continued our discussion on isomorphic groups (Section 9.1 and 9.2).
• Oct. 22, 2018 Midterm 1 has been graded and it will be handed back in class. We started Chapter 9 on isomorphisms.
• Oct. 18, 2018 Today was Midterm 1.
• Oct. 16, 2018 We continued our discussion of cosets and applications of Lagrange's theorem. We used Lagrange's theorem to prove Euler and Fermat's theorems.
• Oct. 15, 2018 Welcome back from the break. We started looking at cosets (Chapter 6). In particular, I proved Lagrange's Theorem.
• Oct. 4, 2018 We looked at Section 5.2 on the dihedral group. The first midterm will cover up to today's material.
• Oct. 2, 2018 We continued with Chapter 5.1. In particular, we discussed odd and even permutations, and I introduced the alternating group.
• Oct. 1, 2018 We started Chapter 5. I talked about permutations, and I discussed cycle notation.
• Sept. 28, 2018 We finished up Chapter 4 on cyclic groups. Today we looked at subgroups of C^*. Please note that I made a small correction to homework assignment 2.
• Sept. 25, 2018 We went over the properties of cyclic groups (Section 4.1).
• Sept. 24, 2018 I introduced cyclic groups (Section 4.1).
• Sept. 21, 2018 In class on Thursday we went over the basic properties of groups. I also introduced subgroups (see Sections 3.2 and 3.3).
• Sept. 19, 2018 Please note that there was a typo in Exercise 5. I have corrected the typo in the exercise.
• Sept. 18, 2018 I formally defined groups and gave a number of examples.
• Sept. 17, 2018 I discussed the set of integers modulo n and the Dihedral Group of order 4 (see Section 3.1).
• Sept. 14, 2018 We finished Chapter 2. I also spent some time discussing how to write proofs (and finding them).
• Sept. 11, 2018 We covered more of Chapter 2 by exploring some of the properties of integers. In particular, I described the Euclidean Algorithm to find the gcd of two numbers.
• Sept. 10, 2018 I started Chapter 2 by reviewing induction. I also introduced the Well-Ordering Axiom, and proved the Division Algorithm.
• Sept. 5, 2018 I finished up the rest of Chapter 1. Please note the classroom will move starting on Monday. We will meet in BSB B136.
• Sept. 4, 2018 Today was our first day of class. I went over the adminstrative details of the course, and I gave a review of set theory and functions (see Section 1.1 and 1.2).
• August 2, 2018 I added a SageMathCell for computations. Take a look below and try it out.
• July 30, 2018 I started to set up the website for Fall 2018.

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Homework

There will be five homework assignments. Assignments must conform to the guidelines in the course outline. Your homework will be graded as follows:

1. For every assignment, 3 or 4 questions will graded in detail (e.g., you are required to write complete mathematical proofs). These questions will be graded out of 5 pts using the rubric described in the course handout.
2. The remaining questions will be graded for completion (1pt each)

Assignments are posted below.

• Assignment 1 (Due: Sept. 20)
• Assignment 2 (Due: Oct. 4)
• Assignment 3 (Due: Oct. 25)
• Assignment 4 (Due: Nov. 8)
• Assignment 5 (Due: Nov. 26) (Note the date! This is a Monday)

For more information on writing proofs, the following notes may help:

• Guide to writing proofs
  A handout on writing proofs I wrote for algebra in 2010.
• Some remarks on writing mathematical proofs
  A nice short guide to writing mathematics by John M. Lee.

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SAGE

As part of this course, the assignments will make use of SAGE, an open-source mathematics program. For the assignments, I will ask you to work through the relevant SAGE tutorials found the book's webpage. You can download your own version of SAGE, or you can try to use CoCalc. Alternatively, you can use the following shell to test your code. This shell allows you to do small calculations.

The following links may help:

• For tutorials, more information on SAGE, and even a free book on SAGE and mathematics, see George Bard's SAGE website
• SAGE's documentation for group theory

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Quizzes

There will be five quizzes. Each quiz will be about 20 minutes, and will be held during tutorial. The dates of the quizzes are

• Quiz 1 - September 14
• Quiz 2 - September 28
• Quiz 3 - October 26
• Quiz 4 - November 9
• Quiz 5 - November 23

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Handouts

All class handouts are available as PDF files.

Course Information
Course handout from first day of class
NOTE: This handout is slightly different than the one given in class. The McMaster policies on this handout have been updated.

Midterm 1 Review Sheet
Handout describing first midterm.

Midterm 2 Review Sheet
Handout describing second midterm.

Exam Review Sheet
Handout describing final exam.

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Grading Scheme

I will calculate your mark using two different weightings. Your final mark will be the higher of the two weights.

Weighting 1
20% = Assignments (5 x 4%)
10% = Quizzes (5 x 2%)
30% = Midterms (2 x 15%)
40% = Final Exam

Weighting 2
20% = Assignments (5 x 4%)
10% = Quizzes (5 x 2%)
15% = Maximum of Midterm 1 and 2
55% = Final Exam

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Important Dates

Sept. 4, 2018
First semester classes begin

Oct. 8-12, 2018
Fall break (no classes)

Oct. 18, 2018
Midterm 1

Nov. 9, 2018
Last day for cancelling courses without failure by default

Nov. 15, 2018
Midterm 2

Dec. 5, 2018
First semester classes end

Dec. 7-20, 2018
Final Exams

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Class Polices

1. Policy on Academic Ethics. You are expected to exhibit honesty and use ethical behaviour in all aspects of the learning process. Academic credentials you earn are rooted in principles of honesty and academic integrity.

Academic dishonesty is to knowingly act or fail to act in a way that results or could result in unearned academic credit or advantage. This behaviour can result in serious consequences, e.g. the grade of zero on an assignment, loss of credit with a notation on the transcript (notation reads: Grade of F assigned for academic dishonesty), and/or suspension or expulsion from the university.

It is your responsibility to understand what constitutes academic dishonesty. For information on the various types of academic dishonesty please refer to the Academic Integrity Policy, located at: http://www.mcmaster.ca/academicintegrity/

The following illustrates only three forms of academic dishonesty:

• plagiarism, e.g. the submission of work that is not one's own or for which other credit has been obtained.
• improper collaboration in group work,
• copying or using unauthorized aids in tests and examinations.

2. Academic Accommodation of Students with Disabilities. Students with disabilities who require academic accommodation must contact Student Accessibility Services (SAS) to make arrangements with a Program Coordinator. Student Accessibility Services can be contacted by phone 905-525-9140 ext. 28652 or e-mail sas@mcmaster.ca. For further information, consult McMaster University's Academic Accommodation of Students with Disabilities policy.

3. Requests for Relief for Missed Academic Term Work. If you have missed work, it is your responsibility to take action. If you are absent from the university for medical and non-medical (personal) situations, lasting fewer than 3 days, you may report your absence, once per term, without documentation, using the McMaster Student Absence Form (MSAF). See Requests for Relief for Missed Academic Term Work

Absences for a longer duration or for other reasons must be reported to your Faculty/Program office, with documentation, and relief from term work may not necessarily be granted. In Math 3GR3, the percentages of the missed work will be transferred to the final examination. Please note that the MSAF may not be used for term work worth 25% or more, nor can it be used for the final examination.

4. Academic Accommodation for Religious, Indigenous or Spiritual Observances (RISO). Students requiring academic accommodation based on religious, indigenous or spiritual observances should follow the procedures set out in the RISO policy. Students requiring a RISO accommodation should submit their request to their Faculty Office normally within 10 working days of the beginning of term in which they anticipate a need for accommodation or to the Registrar's Office prior to their examinations. Students should also contact their instructors as soon as possible to make alternative arrangements for classes, assignments, and tests.

5. Important Message. The instructor and university reserve the right to modify elements of the course during the term. The University reserves the right to change the dates and deadlines for any or all courses in extreme circumstances (e.g., severe weather, labour disruptions, etc.). Changes will be communicated through regular McMaster communication channels, such as McMaster Daily News, A2L and/or McMaster email. If either type of modification becomes necessary, reasonable notice and communication with the students will be given with explanation and the opportunity to comment on changes. It is the responsibility of the student to check their McMaster email and course websites weekly during the term and to note any changes.

6. On-line Statement for Courses Requiring Online Access or Work. In this course we will be using SAGE (which can be accessed online). Students should be aware that, when they access the electronic components of this course, private information such as first and last names, user names for the McMaster e-mail accounts, and program affiliation may become apparent to all other students in the same course. The available information is dependent on the technology used. Continuation in this course will be deemed consent to this disclosure. If you have any questions or concerns about such disclosure please discuss this with the course instructor.

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