## Instructor Contact and General Information

 Instructor: Luís Finotti Office: Ayres Hall 251 Phone: 974-1321 (don't leave messages! -- e-mail (finotti at math dot utk dot edu) me if I don't answer!) e-mail: finotti at math dot utk dot edu Office Hours: by appointment. We can use Blackboard Collaborate (long distance) or you can come to my office.

 Textbook: J. Rotman, "A First Course In Abstract Algebra", 3rd Edition, Prentice Hall, 2006. Prerequisite: One year of calculus or equivalent. Class Meeting Time: Mondays 6pm-7:30pm and Thursdays 2pm-3:30pm via Blackboard Collaborate. (Section 301.) Exams: Midterms: 06/16 (due on Blackboard by 11:59pm). Final: 07/02 (due on Blackboard by 11:59pm). Grade: Best between 34% for HW Average and 33% (each) for Midterm and Final, and 20% for HW Average and 40% (each) for Midterm and Final. (Lowest HW score dropped in the average.) Note the weight of the HWs!

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## Course Information

### Summer Course Warning!

This is a summer course, in which 16 weeks are squeezed into 5. So, as you can imagine, the pace is quite fast. Summer Courses are for very motivated students! If you usually study one hour a day during the regular semester, the equivalent would be to study three hours a day in the summer semester!

You cannot just catch up on the weekends'' in a course like this, as by then we will have covered way too much material. You should catch up immediately if you fall behind, as you will not be able to follow classes and things just start to accumulate in a faster pace than you will likely be able to catch up. I strongly recommend that you review, do problems and study every day!

I've had students taking more than one summer course in the past at the same session, and although it is possible to do it, I'd consider it a Herculean task and would usually advise against it. If you decide to do it, just make sure you are prepared for it! (Tell your loved ones you will see them in July.)

### Course Format

This will be a flipped course, i.e., students will learn a lot on their own, by reading the text and watching short related videos, while the times with the instructor will be spent with questions, solving problems and interactions with students.

You can always request for something you want to see in a video: a problem, some proof in the book, an example, some clarification, etc. If you think it can be done well enough in a lecture (on-line meeting) save it for then, though! If not, just post you request in the "Q&A - Math Related" forum on Blackboard.

Lectures'' will be on-line, via Blackboard, using Collaborate (under Tools -> Collaborate on the left panel of our Blackboard site). I will assign reading and exercises to be done (or attempted) before our lectures. In lecture I will answer questions, solve problems and perhaps provide a few more examples. On the other hand, all (or most of) the content of the lecture will be question driven''. (But, questions such as Can you further explain X?'' or Can you give us examples of Y?'' are more than welcome.) If there are no questions or requests, the lecture will be quite short. It's essential you come to lectures prepared! Otherwise the chances of you getting anything out of this course (and passing it) are quite slim.

I recommend you attend the lectures even if you don't have any questions about the material, as I will take surveys and ask questions that might be relevant to all. You also might learn different ways of doing (or viewing) some problems.

It is only the second time I am teaching a course in this format (flipped/online), so your feedback is quite important and will help shape the course.

### Forums (Discussion Boards)

I've set up four forums on Blackboard: Math Related, Course Structure, Computer Related (questions about Collaborate, LaTeX, Blackboard, etc.) and Feedback (more on this last one below).

I urge you to use these often! If you are ever thinking of sending me an e-mail, think first if it could be posted in these forums. That way my answer might help others that have the same questions as you and will be always available to all. (Of course, if it is something personal (such as your grades), you should e-mail me instead.)

In all these forums you can post anonymously. (Just be careful to check the proper box!) But please don't post anonymously if you don't feel compelled to, as it would help me to know you, individually, much better.

Students can (and should!) reply to and comment on posts on the forum. Discussion is encouraged here! But please be careful with Math Related questions! You should not answer (or ask) questions about how to do a HW problem! (You can ask for hints or suggestions, though.) If you are uncertain if you can answer a (math related) post, please e-mail me first!

Also, make sure to choose the appropriate forum for your question.

Please subscribe to all the forums to be notified of new posts!

### Course Content

Math 506 is a course in basic Abstract Algebra.

We will start by studying the integers (Chapter 1), which are a particular example of commutative rings. We will then move on to (abstract) commutative rings in general (Chapter 3).

Next, we will study the specific case of polynomial rings and its similarities with integers.

Finally, we will study permutations, which are a particular example of groups. We then move on to groups in abstract.

Emphasis will be given to computations, but you can expect a few simple proofs here and there, especially when dealing with abstract rings and groups.

### Chapters and Topics

The goal would be to cover the following sections of our textbook (skipping some parts):

• Chapter 1: Sections 1.3 to 1.5.
• Chapter 3: Sections 3.1 to 3.3 and 3.5 to 3.7.
• Chapter 2: Sections 2.2 to 2.5.

Sections 1.1, 1.2 and 2.1 are prerequisites.

We will skip quite a few parts from the sections above. For more details on what we will skip, check the description of the lectures, where I specify what you need to read for each lecture.

Other topics (and digressions) might also be squeezed in as time allows.

### Homework Policy

Homeworks and due dates are posted at the section Reading and Homework of this page. (They should also appear in your Blackboard Calendar.)

Homeworks must be turned in via Blackboard. (Please, don't e-mail them to me unless strictly necessary, e.g., Blackboard is not working.) Just click on Assignments (Submit HW)'' on the left panel of Blackboard and select the correct assignment.

Scanned copies are acceptable, but typed in solutions are preferred. I recommend you learn and use LaTeX. (Resources are provided below.)

Points might be taken from messy solutions in all assignments, and you need to show work in all questions (unless stated otherwise)! This same applies to HWs, exams and all graded work.

In my opinion, doing the HW is one of the most important parts of the learning process, so the weight for them is quite high.

Also, you should make appointments for office hours having difficulties with the course. I will do my best to help you. Please try to first ask questions during class time (online)! I will not take appointments from students who don't attend the lectures (unless there is a good excuse, of course).

Finally, you can check all your scores at Blackboard.

### E-Mail Policy

I will assume you check your e-mail at least once a day, but preferably you should check your e-mail often. I will use your e-mail (given to me by the registrar's office) to make announcements. (If that is not your preferred address, please make sure to forward your university e-mail to it!) I will assume that any message that I sent via e-mail will be read in less than twenty four hours, and it will be considered an official communication.

Moreover, you should receive e-mails when announcements are posted on Blackboard, or where there is a new post in any of our forums. (Again, please subscribe to all of them, to receive notifications! Important information my appear in those.)

### Feedback

Please, post all comments and suggestions regarding the course using Blackboard's Feedback. These can be posted anonymously (or not), just make sure to check the appropriate option. Others students and myself will be able to respond and comment. If you prefer to keep the conversation private (between us), you can send me an e-mail, but then, of course, it won't be anonymous.

### Introductions

You are invited to post an introduction about yourself on Blackboard. (There is an Introductions'' link.) This is not required at all! But it might make things a bit more social.

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## Legal Issues

### Conduct

All students should be familiar and maintain their Academic Integrity: from Hilltopics, pg. 46:

The university expects that all academic work will provide an honest reflection of the knowledge and abilities of both students and faculty. Cheating, plagiarism, fabrication of data, providing unauthorized help, and other acts of academic dishonesty are abhorrent to the purposes for which the university exists. In support of its commitment to academic integrity, the university has adopted an Honor Statement.

All students should follow the Honor Statement: from Hilltopics, pg. 16:

Honor Statement

"An essential feature of The University of Tennessee is a commitment to maintaining an atmosphere of intellectual integrity and academic honesty. As a student of the University, I pledge that I will neither knowingly give nor receive any inappropriate assistance in academic work, thus affirming my own personal commitment to honor and integrity."

You should also be familiar with the Classroom Behavior Expectations.

We are in a honor system in this course!

### Disabilities

Students with disabilities that need special accommodations should contact the Office of Disability Services and bring me the appropriate letter/forms.

### Sexual Harassment and Discrimination

For Sexual Harassment and Discrimination information, please visit the Office of Equity and Diversity.

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## Course Goals and Outcomes

### Course Relevance

This is a first (and basic) course in an important area of mathematics: Algebra. It's an important topic, and although quite abstract at times, it has many applications in real life, such as in cryptography, error-correcting codes and even Google's seach algorithm.

### Course Value

The students will:
• develop analytic and critical thinking (as in any math course);
• get a deeper understanding of fundamental mathematical concepts, such as integers and polynomials;
• develop a deeper abstraction level.

### Student Learning Outcomes

At the end of the semester students should be able to:
• perform basic computations with integers and polynomials (long division, GCD, LCM, congruences);
• prove basic facts about integers and polynomials;
• understand and recognize basic algebraic structures, such as groups, fields and rings;
• prove basic properties of these algebraic structures.

### Learning Environment

• Type: This will be a flipped course, i.e., students will learn a lot on their own, by reading the text and watching short related videos, while the times with the instructor will be spent with questions, solving problems and interactions with students.
• Where: Students will work from home in activities such as reading, watching video, participating in video conferences and long distance office hours. A lot of the discussions should happen on the forums on Blackboard.
• Student and Faculty roles:
• Students will have to be more active in the learning process than in regular courses, as they will do most of the reading and learning on their own.
• The instructor will be a facilitator, answering questions and offering advice and guidelines, answering questions and providing feedback.
• Students Responsibilities:
• Keep up with the schedule, i.e., read the assigned sections, watch the recommended videos and solve assigned problems according to the schedule. This is crucial in this flipped format!
• Carefully work on assigned problems.
• Carefully review graded work to learn from past mistakes.
• Check the course site often (at the very least once a day) for assignments and announcements.
• Search for help if having difficulties!
• Provide feedback to improve the course.
• Instructor Responsibilities:
• Be available for help.
• Provide examples and solve problems.
• Be open to discussions concerning content, format and evaluations.
• Provide relevant problems and exercises for homework, quizzes and exams.
• Provide feedback to the students.

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Here are some other books you might find helpful:

The first two books are considered "easier" books. The Artin's book is of a bit higher level (and has a slightly different focus). The last one is a "standard" text for a first course in abstract algebra, but have a higher level of difficulty than the previous two. It's been used for the honors section of the undergraduate algebra course here at UT.

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## LaTeX

LaTeX is the most used software to produce mathematics texts. It is quite powerful and the final result is, when properly used, outstanding! Virtually all professional math text you will ever see is done with LaTeX, or one of its variants.

LaTeX is available for all platforms and freely available.

The problem is that it has a steep learning curve at first, but after the first difficulties are overcome, it is not bad at all.

One of the first difficulties one encounters is that it is not WYSIWYG (what you see is what you get''). It resembles a programming language: you first type some code and then this code is processed to produce a nice document (a non-editable PDF file, for example). Thus, one has to learn how to code'' in LaTeX, but this brings many benefits.

I recommend that anyone with any serious interest in producing math texts to learn it! On the other hand, I don't expect all of you to do so. But note that there are processors that can make it easier'' to create LaTeX documents, by making it point-and-click'' and (somewhat) WYSIWYG.

Here are some that you can use online (no need to install anything and files are available online, but need to register):

We will use the first one, SageMathCloud in our course, so you have to register for it, and thus might as well use it. It is probably the best of the services anyway, and it can do a lot more than just LaTeX. You should have received, by the first day of classes, an invitation to collaborate on a project that I've created for this course (Math 506 -- Summer 2015).

If you want to install LaTeX in your computer (so that you don't need an Internet connection), check here.

I might need to use some LaTeX symbols when writing in our online meetings, but it should be relatively easy to follow. I will also provide samples and templates that should make it much easier for you to start.

A few resources:

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## Solutions to Selected HW Problems

Please read: I will try to post here a few solutions. The new solutions will be added to this same file. They might come with no explanation, just the answer''. If yours do not match mine, you can try to figure out again. (Also, read the disclaimer below!) You can come to office hours or ask in class if you want explanations for the answers. Be careful that just because our answers'' were the same, it doesn't mean that you solved the problem correctly (it might have been a fortunate'' coincidence), and in the exams what matters is the solution itself. I will do my best to post somewhat detailed solutions to the harder problems, though.

Disclaimer: I will have to put these solutions together rather quickly, so they are subject to typos and conceptual mistakes. (I expect you to be a lot more careful when doing your HW than I when preparing these.) You can contact me if you think that there is something wrong and I will fix the file if you are correct.

Solutions to Selected HW Problems (Click on Refresh'' or Reload'' if you don't see the changes!)

Note: This file and its LaTeX source are also available in our shared project in SageMathCloud, under the name Solutions.tex.

CHANGE LOG:

• 06/30 -- 1:10am Solutions for Sections 3.5, 3.7 and 2.2-4.
• 06/25 -- 4:30pm Solutions for Sections 3.2 and 3.3.
• 06/22 -- 5:55pm Solutions for Section 3.1.
• 06/12 -- 10:45am Solutions for Sections 1.3-5.

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### Lecture 1: 06/01 from 6pm to 7:30pm

Reading: Course Info, Sections 1.1, 1.2 and 2.1 (briefly).

### Lecture 2: 06/02 from 2pm to 3:30pm

Reading: Sections 1.3 up to Proposition 1.34 on pg. 40.

### Homework 1: 06/09 by 11:59pm

 Section 1.3: Turn in: 1.46(vii), 1.50, 1.53 with $b=3$, 1.55(i). Extra Problems: 1.46, 1.50, 1.52, 1.53, 1.55, 1.57.

### Homework 2: 06/12 by 11:59pm

 Section 1.4: Turn in: 1.68(ii), 1.69(i), 1.76(ii). Extra Problems: 1.68, 1.69, 1.70(i), 1.76(ii).

 Section 1.5: Turn in: 1.77(viii), 1.78(iii), (v), (vi), 1.91(ii). Extra Problems: 1.77, 1.78, 1.79, 1.80, 1.81, 1.83, 1.85, 1.86, 1.91, 1.95.

### Midterm: 06/16 by 11:59pm

Sections: 1.3, 1.4 and 1.5.

### Homework 3: 06/19 by 11:59pm

 Section 3.1: Turn in: 3.1(vi), 3.3(ii), 3.15(i). Extra Problems: 3.1 except (v) and (viii), 3.2, 3.3, 3.6, 3.8(i), 3.13, 3.15(i), (ii).

### Lecture 8: 06/22 from 6pm to 7:30pm

Reading: Sections 3.5, 3.6 and 3.7.

### Homework 4: 06/23 by 11:59pm

 Section 3.2: Turn in: 3.17(ii), (iii), 3.23, 3.27(ii). Extra Problems: 3.17, 3.19, 3.20 (hint: in a domain, if $a \neq 0$ and $ax=ay$, then $x=y$; use that to show that if $a \neq 0$ and $R$ is a finite domain, then $\{ ax \; : \; x \in R\} = R$; use that to show $a$ is a unit), 3.23, 3.27(i), (ii).

 Section 3.3: Turn in: 3.29(ii), 3.30. Extra Problems: 3.29, except (i), 3.30, 3.32, 3.37 (this one should be after 3.5).

### Homework 5: 06/26 by 11:59pm

 Section 3.5: Turn in: 3.56(v), 3.58, 3.62. Extra Problems: 3.56 from (i) to (vii), 3.58, 3.62, 3.64.

 Section 3.7: Turn in: 3.86 (vi), 3.87 (v), (ix). Extra Problems: 3.86, 3.87 except (vii).

### Homework 6: 06/29 (Monday!) by 11:59pm

 Section 2.2: Turn in: 2.21(x), 2.34 (note that $\alpha \beta = \beta \alpha$ iff $\alpha \beta \alpha^{-1} = \beta$) and this question on permutations. Extra Problems: 2.21 ((ii) is easier after 2.3), 2.22, 2.25, 2.26, 2.34, this question on permutations.

### Homework 7: 07/01 - NOT TO BE TURNED IN!

 Section 2.3: 2.36 (i) to (v) and (viii) to (ix), 2.37, 2.38, 2.40.

 Section 2.4: 2.52 (i) to (v) and (x) to (xi), 2.54, 2.55, 2.57.

### Final: 07/02 by 11:59pm

Chapters 2 and 3. (No Chapter 1.)

If the due date for an assignment is close but it is still marked as tentative'', post a request to update it on the Q&A - Course Structure'' forum on Blackboard, and I'll update it and let you know.

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