Course Info

Title: MAT 612: Abstract Algebra II
Semester: Spring 2016
Credits: 3
Section: 1
Time: MWF at 9:10-10:00AM
Location: AMB 207

Instructor Info

  Dana C. Ernst, PhD
  AMB 176
  11:15-12:15 MWF and 9-10 TTh (or by appointment)


MAT 511 with grade of C or better.

Catalog Description

Continuation of MAT 511. Rings and modules, field extensions and Galois theory, and advanced topics at instructor’s discretion. Letter grade only.


There is no required textbook. All necessary material (including homework) will be made available via handouts and postings on the course webpage. However, I will be loosely following “Abstract Algebra” by Dummit and Foote. So, if you want to purchase a textbook to use as a resource, this is the book I recommend getting. I’ve also listed a few free abstract algebra books on the Course Materials page. Regardless of what resource you decide to use, the only way to achieve a sufficient understanding of the material is to be digesting it in a meaningful way. You should be seeking clarification about the material whenever necessary by asking questions in class, working with our students, stopping by office hours, or emailing me.

Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

Outline of Course

The course will consist of four parts. I’m not sure what order will cover these topics.

  1. Ring Theory. We will pick up where we left off in MAT 511 and continue our study of rings. In particular, we introduce Euclidean domains and unique factorization domains (UFDs) and continue to tinker with principal ideal domains (PIDs) and polynomial rings.
  2. Module Theory. After introducing basic definitions and examples, we study module homomorphisms and quotients. We will also take a look at some subset of free modules, direct sums of modules, and exact sequences involving modules.
  3. Field theory and Galois Theory. We will take a closer look at fields (including finite fields) and introduce field extensions. Time permitting, we will examine classical straightedge and compass constructions. Arguably, the punchline of the course is the Fundamental Theorem of Galois Theory. In addition to proving this theorem, we will study several applications including the computation of Galois groups, solvable and radical extensions, and insolvability of the quintic.
  4. Group Theory. We will dabble in a few additional group theory topics including nilpotent groups, solvable groups, and semi-direct products. Time permitting, we will work on classifying groups of small order.

Learning Management System

We will make limited use of Bb Learn this semester, which is Northern Arizona University’s default learning management system (LMS). Most course content (e.g., syllabus, course notes, homework, etc.) will be housed here on our course webpage that lives outside of Bb Learn.


You are allowed and encouraged to work together on homework. Yet, each student is expected to turn in their own work. In general, late homework will not be accepted. However, you are allowed to turn in one late homework assignment with no questions asked. Unless you have made arrangements in advance with me, homework turned in after class will be considered late.

Generally, the written homework assignments will be due on Wednesdays, but I will always tell you when a given homework assignment is due–so there should never be any confusion. Your homework will always be graded for completion and some subset of the problems will be graded for correctness. Problems that are graded for completeness will be worth 1 point. Problems that are graded for correctness will either be worth 2 points or 4 points depending on the level of difficulty. Generally, computational problems will be worth 2 points while problems requiring a formal proof will be worth 4 points. Each 4-point problem is subject to the following rubric:

Grade Criteria
4 This is correct and well-written mathematics!
3 This is a good piece of work, yet there are some mathematical errors or some writing errors that need addressing.
2 There is some good intuition here, but there is at least one serious flaw.
1 I don't understand this, but I see that you have worked on it; come see me!
0 I believe that you have not worked on this problem enough or you didn't submit any work.

To compute your score on a given homework assignment, I will divide your total points by the total possible points to obtain a percent score. Each homework assignment has the same weight. Your overall homework grade will be worth 25% of your final grade. If you utilize any resources other than our course notes or Dummit and Foote, I expect you to cite your sources. Better yet, try to complete your homework without relying on external resources.


There will be 3 midterm exams and a cumulative final exam. Exam 1 and Exam 2 will be written exams and may include a take-home portion. These exams are tentatively scheduled for Friday, February 26 and Friday, April 15, respectively, and each is worth 20% of your overall grade. Exams 1 and 2 are likely to have take-home components. Exam 3 will be an oral exam taken individually in my office. The questions for the oral exam will predominately come from homework problems. Exam 3 will last roughly 30 minutes. Your score on Exam 3 together with your performance on the presentation of in-class theorems (see below) will be worth 10% of your overall grade. Students will schedule Exam 3 during April 25-May 12. The final exam will be on Wednesday, May 11 at 7:30-9:30AM and is worth 25% of your overall grade. Make-up exams will only be given under extreme circumstances, as judged by me. In general, it will be best to communicate conflicts ahead of time.


A couple times each week, I will ask individuals to prepare the proof of a theorem, or possibly a detailed description of an example, that will be presented to the class. You are allowed to consult outside resources when preparing for these presentations (but you should cite your sources). The goal will be for each student to present twice. All presentations will be graded using the rubric below.

Grade Criteria
4 Completely correct and clear proof or solution. Yay!
3 Proof has minor technical flaws, some unclear language, or lacking some details. Essentially correct.
2 A partial explanation or proof is provided but a significant gap still exists to reach a full solution or proof.
1 Minimal progress has been made that includes relevant information & could lead to a proof or solution.
0 You were completely unprepared.

Your overall performance during presentations together with your score on Exam 3, will be worth 10% of your overall grade.

Basis for Evaluation

In summary, your final grade will be determined by your scores in the following categories.

Category Weight
Homework 25%
Exam 1 (written) 20%
Exam 2 (written) 20%
Exam 3 (oral) and Presentations 10%
Final Exam 25%

Grade Determination

In general, you should expect the grades to adhere to the standard letter-grade cutoffs:

A 100-90%, B 80-89%, C 70-79%, D 60-69%, F 0-59%.

Department and University Policies

You are responsible for knowing and following the Department of Mathematics and Statistics Policies (PDF) and other University policies listed here (PDF). More policies can be found in other university documents, especially the NAU Student Handbook (see appendices) and the website of the Office of Student Life.

As per Department Policy, cell phones, MP3 players and portable electronic communication devices, including but not limited to smart phones, cameras and recording devices, must be turned off and inaccessible during in-class tests. Any violation of this policy will be treated as academic dishonesty.

Class Etiquette

Students are expected to treat each other with respect. Students are also expected to promote a healthy learning environment, as well as minimize distracting behaviors. In particular, you should be supportive of other students while they are making presentations. Moreover, every attempt should be made to arrive to class on time. If you must arrive late or leave early, please do not disrupt class.

Please turn off the ringer on your cell phone. I do not have a strict policy on the use of laptops, tablets, and cell phones. You are expected to be paying attention and engaging in class discussions. If your cell phone, etc. is interfering with your ability (or that of another student) to do this, then put it away, or I will ask you to put it away.

Important Dates

Here are some important dates:

  • Thursday, January 28: Last day to drop/add (no W appears on transcript)
  • Friday, February, 26: Exam 1
  • Monday, March 14-Friday, March 18: Spring Break!
  • Friday, March 25: Last day to withdraw from a course (W appears on transcript)
  • Friday, April 15: Exam 2
  • April 25-May 12: Exam 3
  • Wednesday, May 11: Final Exam

Changes to the Syllabus

Any changes in this syllabus made during the term will be properly communicated to the class.

Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
  Google Scholar
  Impact Story

Current Courses

  MAT 411: Abstract Algebra
  MAT 690: Genome Combinatorics

About This Site

  This website was created using GitHub Pages and Jekyll together with Twitter Bootstrap.

  Unless stated otherwise, content on this site is licensed under a Creative Commons Attribution-Share Alike 4.0 International License.

  The views expressed on this site are my own and are not necessarily shared by my employer Northern Arizona University.

  The source code is on GitHub.

Land Acknowledgement

  Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.