Navigation
- Zoom Meeting Links:
- Class: See the Syllabus on Canvas.
- Office Hours: See the Syllabus on Canvas.
- Homework Problems
- Canvas: important announcements, grades, calendar, etc.
- Piazza (Discussion Boards).
- Instructor Contact and General Info:
- Course Description and Information:
- Zoom -- NEW!
- Legal Issues:
- Additional Bibliography
- LaTeX
- Links
- Videos
- Handouts
Instructor Contact and General Information
Instructor: | Luís Finotti |
Office: | Ayres Hall 251 |
Phone: | 974-1321 (don't leave messages! -- e-mail me if I don't answer!) |
e-mail: | lfinotti@utk.edu |
Office Hours: | MW 11-12, or by appointment via Zoom. |
Textbook: | J. Rotman, "A First Course In Abstract Algebra", 3rd Edition, Prentice Hall, 2006. |
Prerequisite: | Math 300/307 (and 251/257). |
Class Meeting Time: | MWF 10:10-11 |
Exams: | Midterm 1 (Section 1.3): 02/05 (Wed). |
Midterm 2 (Sections 1.4, 1.5): 03/06 (Fri). | |
Midterm 3 (Sections 3.1, 3.2): 03/25 (Wed). | |
Midterm 4 (Sections 3.3, 3.5, 3.6): 04/15 (Wed). | |
Final: 05/01 (Friday) 8-10. | |
Grade: | 20% for each midterm with lowest dropped + 40% for the final. |
See here for letter grade ranges. |
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Course Information
Course Content
This course is a one-semester introduction to Abstract Algebra. Emphasis will be given to integers and polynomials, which are examples of commutative rings. The other main topic to be covered (at least superficially) is groups.
This course might be a bit of a shock to many students, as up to now most will not have dealt with discrete, rather than continuous (in the calculus sense) structures and proofs, which is what you usually deal with in calculus, differential equations, and when working with real numbers. So, it might take a little time for you to get use to the ideas and techniques used in this course.
Being an upper level course for math majors, most of the course will be spent on proofs (as in Math 300/307), and you will have to read and write many. I will assume you are comfortable doing both. We will also deal with induction and set theory (again from Math 300/307.) Other than that, there is really very little in terms of background knowledge necessary, except for matrices (Math 251/257), which will be used as examples.
Chapters and Topics
The goal would be to cover the following sections of our textbook (skipping some parts):
- Chapter 1:
- Sections 1.3, 1.4: All.
- Section 1.5: We will skip Example 1.78 (on pg. 70) until the end of the section. (It's quite interesting, but we don't have time.).
- Chapter 3:
- Sections 3.1, 3.2: All.
- Section 3.3: The text gives a formal construction of polynomials here, but I will skip it and just treat them as the familiar objects that they are (or seem to be). Other than that, we will cover the whole section.
- Section 3.5: We will skip from Corollary 3.54 (pg. 259) to Theorem 3.63 (inclusive, on pg. 262) and from the subsection on Euclidean Rings (on pg. 267) until the end of the section.
- Section 3.6: We will skip from Lemma 3.87 (on pg. 278) to the end of the section.
- Section 3.7: If we are pressed on time, we might skip this section altogether (to get to Groups). If we do cover it, we will likely cover it all.
- Chapter 2:
- Section 2.2: All.
- Section 2.3: We will skip the subsection Symmetries (on pg. 137) to the end of the section.
- Section 2.4: If time allows, we should cover it all.
Sections 1.1, 1.2 and 2.1 are prerequisites. On Section 1.2 you can skip all that comes after Corollary 1.26 (on pg. 27), though.
Although not very likely, this outline is subject to change!
Homework Policy
Homework problems are posted below. As soon as we finish a section in class, you should start working on the problems from the section in the list. For longer sections, you should start before we finish it. Just look for problems that have statements on the topics we have already covered.
On the other hand, HW will not be collected or graded! (Also, there are no quizzes.) The point of the HW is to learn and practice for the exams. In my opinion, doing the HW is one of the most important parts of the learning process, so even if it does not count towards your grade, I recommend you take it very seriously.
Solutions to the HW will be posted on Canvas and you can bring your questions to class. In particular, I will try to set sometime to answer HW questions the class before each exam.
Note that you should not look at the solutions before you try, for a considerable amount of time, to do the problems! But you should always check your solution against the posted one and bring questions to class (or post on Piazza).
You have to be extremely careful with this set up. If you are not responsible and motivated, and end up not taking your HW seriously (even though it is not collected), your chances of passing this course are quite slim! But, then again, you are adults, so I don't feel guilty asking you to be responsible.
Also, you should make appointments for office hours having difficulties with the HW or the course in general! (Even if it is not for grade, you have to do well on your HW!) I will do my best to help you.
Extra Practice Problems
A common question (with good reason) for this course is what extra problems students can do to get more practice. Here are some suggestions:
- It goes without saying, but do all homework problems.
- I often leave some facts as exercises in class. Do all of those. (Some of those are quite easy, and, if you are 100% sure you know how to do them, then you can skip them.)
- A lot of the things I prove in class are of the same kind as the ones that show up in HW or exams. Try redoing some of those (without looking or reviewing the proof, of course). Just be careful that a few of the things I prove are not that easy.
- Look for problems from my old exams. You can find links to my old courses in the section Links below. (Remember, you can always ask me if a problem is relevant to our course (or to a particular exam) if in doubt.)
- If the problems are computational, you can try to "change the numbers" from assigned problems to make new problems.
- Do extra problems from our text (besides the ones on the HW). You have to be careful if we skipped something from a section, but, again, you can always check with me.
- If you are preparing for an exam, you can redo problems that you did a while before, especially those with which you had difficulty. (If you forgot it, it's like doing a new problem.)
- You can also do problems from other textbooks I list in Additional Bibliography, especially the first two. Again, this can be a bit problematic, as maybe they cover something we didn't (or vice-versa, and hence have no problems for a particular sub-topic), but, also again, you can always ask me. Except for Chapter 1, you should be able to find the other topics (on Rings and Groups) from our course in those references, and their corresponding exercises. The library might have some copies. (Don't worry about editions.)
Statements and Index Cards
I strongly recommend you write in a separate sheet of paper all definitions and statements of important theorems (lemmas, corollaries, propositions, etc.), and perhaps even a few more important examples that illustrate some technique. I recommend you do it before you start your HW on the corresponding section!
There are two main reasons for doing so: firstly, the act of writing helps you review and remember the main tools to solve problems in your HW. Secondly, having them on a separate sheet of paper makes it quicker to find what you need when doing your HW. (Hopefully by now you are aware that it is impossible to solve problems without knowing the relevant definitions and theorems!)
I would also recommend you write the definitions and theorems covered in class before the following class. This will help you follow better the new lecture. In fact, there is some benefit in writing them before they are introduced in class, as it makes easier to follow that lecture. But, in any event, you should do it before you do your HW.
Also, you will be allowed to bring two index cards for each midterm and four for the final. If you have the statements already in a sheet of paper, and studied with it, you will probably know which ones you should put in your index cards.
These index cards must be 3" by 5" (you can cut a piece of paper to that size, though) and have your name in each one of them, but you can write on both sides and put whatever you want in them. (You are not allowed to share or exchange cards during the exams!)
I was reluctant on allowing the use of index cards, since in my opinion you should study enough to know these definitions and theorems. But I also believe that it does help writing them: you have to look over all the statements and assess which are most important and write them again. Also, it allows you to spend more of your time, when preparing for exams, on the most important thing: solving problems!
Another warning: don't rely too much on these cards. Having the statements are not enough to know how to use them! It would take too much time for you to figure everything out from them. You need practice using them!
How to Be Successful in this Course
- Study hard!
- Write statements of main theorems and results for quick reference (and to help you memorize them).
- Review the material before classes.
- Work on all the HW problems. Don’t look at the solutions until you’ve tried for a while. You will only learn by working on problems!
- Don’t let a HW problem "pass". You should always try to find how to solve every problem.
- Look for help if you are having trouble: post questions on Piazza (web forum) or come see me.
- If you can’t do a problem and do get help on it:
- Look for what you were missing! (Did you forget a theorem? Were you missing a particular idea?) Seeing the solution won’t help if you don’t get anything out of it .
- Go back to the problem a couple of days later and redo it by yourself.
- Ask questions in class.
- Work on old exams to prepare for our exams.
- Watch the videos posted below.
Piazza (Discussion Board)
We will use Piazza for online discussions. The advantage of Piazza (over other discussion boards) is that it allows us (or simply me) to use math symbols efficiently and with good looking results (unlike Canvas). It also allows anonymous posts (also unlike Canvas).
To enter math, you can use LaTeX code. (See the section on LaTeX below.) The only difference is that you must surround the math code with double dollar signs ($$) instead of single ones ($). Even if you don't take advantage of this, I can use making it easier for you to read the answers.
You can access Piazza through the link on the left panel of Canvas or directly here: https://piazza.com/utk/spring2020/math351/home. (There is also a link at the "Navigation" section on the top of this page and on the Links section.)
To keep things organized, I've set up a few different folders/labels for our discussions:
- Chapters and Exams: Each chapter and exam has its own folder. Ask question related to each chapter or exam in the corresponding folder.
- Course Structure: Ask questions about the class, such as "how is the graded computed", "when is the final", etc. in this folder. (Please read the Syllabus first, though!)
- Computers: Ask questions about the usage of LaTeX, Piazza itself and Canvas using this folder.
- Feedback: Give (possibly anonymous) feedback about the course using this folder.
- Other: In the unlikely event that your question/discussion doesn't fit in any of the above, please use this folder.
I urge you to use Piazza often for discussions! (This is specially true for Feedback!) If you are ever thinking of sending me an e-mail, think first if it could be posted there. 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.)
Note that 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 Piazza. Discussion is encouraged here!
Also, please don't forget to choose the appropriate folder(s) (you can choose more than one, like a label) for your question. And make sure to choose between Question, Note or Poll.
When replying/commenting/contributing to a discussion, please do so in the appropriate place. If it is an answer to the question, use the Answer area. (Note: The answer area for students can be edited by other students. The idea is to be a collaborative answer. Only one answer will be presented for students and one from the instructor. So, if you want to contribute to answer already posted, just edit it.) You can also post a Follow Up discussion instead of (or besides) an answer. There can be multiple follow ups, but don't start a new one if it is the same discussion.
Important: Make sure you set your "Notifications Settings" on Piazza to receive notifications for all posts: Click on the gear on the top right of the Piazza site, the choose "Account/Email Setting", then "Edit Email Notifications" and then check "Automatically follow every question and note". Preferably, also set "Real Time" for both new and updates to questions and notes. I will consider a post in Piazza official communication in this course, I will assume all have read every single post there!
You can also use Piazza for Private Messages. I'd prefer you use e-mail to talk to me, unless it is a math question (in which either you or I would need to enter math symbols) that cannot be posted for all (such as an exam question). You can also send private messages to fellow students, but keep in mind that I can see those too! (So, not really that private...)
You should receive an invitation to join our class in Piazza via your "@tennessee.edu" e-mail address before classes start. If you don't, you can sign up here: https://piazza.com/utk/spring2020/math351. If you've register with a different e-mail (e.g., @vols.utk.edu) you do not need to register again, but you can consolidate your different e-mails (like @vols.utk.edu and @tennessee.edu) in Piazza, so that it knows it is the same person. (Only if you want to! It is recommended but not required as long as you have access to our course there!) Just click on the gear icon on the top right of Piazza, beside your name, and select "Account/Email Settings". Then, in "Other Emails" add the new ones.
Missed Work
Unless other arrangements are made beforehand, missed midterms, with documented excuses (see below), will be made up by the part of the (comprehensive) final that corresponds to the missed midterm.
More precisely: say you missed Midterm 4, which involved sections 2.2 and 2.3, and say that in our final questions 3 and 4 (and only those) are about the material of those sections. Then, the points you get in those questions of the final will make you Midterm 4 grade.
Your justification for missing an exam has to be processed by the Office of the Dean of Students, more precisely, here. Note that, as stated in the referred site, final approval of all absences and missed work is determined by the instructor. (So, just because you've submitted a justification through the Office of the Dean of Students, it doesn't mean it will be accepted by me.)
Communications and E-Mail Policy
You are required to set up notifications for Piazza (as explained above) and for Canvas to be sent to you immediately. For Canvas, check this page and/or this video on how to set your notifications. Set notifications for Announcements to "right away"! (Basically: click on the the profile button on left, under UT's "T", then click "Notifications". Click on the check mark ("notify me right away") for Announcements.)
Moreover, I may send e-mails with important information directly to you. I will use the e-mail given to me by the registrar and set up automatically in Canvas. (If that is not your preferred address, please make sure to forward your university e-mail to it!)
All three (notifications from Piazza, notifications from Canvas and e-mails) are official communications for this course and it's your responsibility to check them often!
Feedback
Please, post all comments and suggestions regarding the course using Piazza. Usually these should be posted as Notes and put in the Feedback folder/label (and add other labels if relevant). 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 (not anonymous), or a private message in Piazza (possibly anonymous).
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Zoom
Using Zoom
Starting on Monday 03/23 our classes and office hours will all be through Zoom. (See the links in Syllabus on Canvas.)
Before our first lecture on 03/23, please familiarize yourself with Zoom. At the very least, read the Getting Started and Best Practices for Participants pages from OIT. See also Remote It: Just for Students.
Please login to your account and test the microphone and camera before class.
Also, please make sure you have a way to scan documents quickly (e.g., from your phone) to post your online exams. (See Section Online Exams below.)
I will be online on Zoom at around 9:45am on our first day (03/23) in case you want to login early and check if everything is working, but the actual class starts at 10:10 as usual.
Lectures
The classes will be live, so you should plan to be "present" during those meetings. On the other hand, the classes will be recorded and posted on Canvas for later viewing or in case you had to miss it.
Instead of writing on the board, I will go over prepared notes. These will be available on Canvas ahead of time, so you can download them and see them in your own computer. Revised notes might be posted (also in Canvas) if we found any mistakes or problems.
For questions you asked during lectures (for which I don't have prepared notes), I will type an answer live (using LaTeX). I am not very fast at all typing, so this might be a bit slow. I will type just enough that, together with my explanations, all can follow. After class I will edit this document (to make it more readable) and post it as a PDF in Canvas for future reference.
The notes, videos, and PDF for questions can be found in Canvas by (after selecting our course) clicking on "Pages" on the left panel and then on the link for the date's lecture.
Office Hours
Office Hours will also be via Zoom.
Note that office hours will not be recorded! You can record it yourself if you want to or request me to do it, but I will not do it by default (unlike our lectures).
Remember that our office hours are MW (not Fridays) from 11 to 12. I should be logged in Zoom during those times, but keep in mind I may walk away for a bit if no one is there. I should not be away for long, so you can wait a little. If you don't see me, feel free to turn on your microphone and ask for me. (In case I fall asleep.) But please be a little patient if I don't respond, as I should be back soon.
Although not necessary in principle, if you let me know you will come to office hours (and at what time exactly), I can make sure to not be away at that time.
Also, I can take appointments for office hours at different times if necessary. In that case, please write me to make an appointment, but use the same Zoom link as for regular office hours.
Keep in mind that these office hours are shared with another course I'm teaching.
During Lectures
Here are some notes about how we will handle the lectures:
- Please, keep your microphones on mute! If you have a question, please raise your hand. You can turn on your mic and ask your question after I call you. Please do not forget to turn it off after I answer your questions.
- You can also ask questions via chat. (These can be private, i.e., to just me.) I've missed some of those in the past (I'm not sure I get a sound notification when I get a message), but I will try to be more careful this time. But, if I don't answer a question you sent via chat, please raise your hand.
- If possible, please turn on your cameras. It helps me to see your faces to see if you look confused or content. It's very hard to "read the room" in Zoom. But you certainly are not required to have your camera on.
- Important: For the same reason above, I beg you to please use non verbal feedback as often as possible! This will help me have a better idea if I need to explain something more carefully or slow down/speed up. Please use it!
Online Exams
I decided to try to keep things as simple as possible. In order to do that, I need to trust that you will follow our code of conduct, as made explicit in the section Legal Issues. (Please, don't make me regret this.) Anyone who is caught cheating (in any way) will be reported and will receive zero for the exam! Moreover, if I suspect there was cheating, we will move to a much more elaborate system that will require, among other things, two devices recording the exam and using/installing LockDown Browser.
The idea is to keep the exam exactly as before, including the index cards and questions taken from HW.
The exam will be during class time, also using Zoom. You will all be required to have your camera on at all times and with an angle I can see you clearly.
Keep your sound on, in case I need to send you or the class a chat message, or if I need to turn on my microphone to say something to the whole class. (Note: You need to keep your sound on, not your microphone, as the point is that you hear me if I need to talk to you.)
Be prepared to, at any time upon request, show your room/environment to the camera and/or share your screen(s)! I will likely do some random checks, or might ask you if I notice anything suspicious. I will do this in a private Zoom Break Room to not disturb the rest of the class. Delaying to comply will be considered cheating, and therefore will give you a zero for the exam!
At exactly 10:10 (or as soon as all students are logged in), I will send a link to the exam via chat message. Students who come late will need to send me a private message asking for the link.
In your solutions, you do not need to copy the questions statements, just number them clearly. Also, please use a different page for each question!
If you have any questions during the exam, you can send me a private message with your question. (You can only use your computer to look at the exam or to send me private messages! If see you typing in your computer and don't receive a private message, I will have to assume you were cheating!)
Once you are done with the exam, you need to send me a private message saying that you will start scanning/uploading. (Something short, like "Scanning now", works.) You will have to upload your solutions to Canvas. You can do it by clicking on "Assignments" on the left panel of Canvas, and then selecting the corresponding exam.
Please, upload, if at all possible, a single PDF with all the pages. (A little more on this below.) Please do not just upload pictures for each page (unless you absolutely have to). Also, please don't e-mail me your solutions unless you have problems uploading it to Canvas. The scans have to be perfectly legible!
Since you have to deal with scanning and uploading, I will give you some extra time to do so: you will need to have your exam uploaded to Canvas by 11:30!
You can only leave the Zoom meeting after you upload your exam. You will then send me a private message, like "Done." Don't forget this step! I will match the time I get your message (and you left Zoom) with the time you've uploaded the exam. If these don't match, you will get zero!
Again, you cannot leave Zoom or turn off you camera until you upload your exam to Canvas and send me a private message saying you are done.
The exams will be graded and returned through Canvas, and you will be able to see my comments in your graded exam.
Scanning
Below are some apps that can scan documents from your phone. Of course, it is not an exhaustive list and you can use anything that works well. Most of these I have no experience using, so don't consider them as a real endorsement in anyway. Especially the iOS ones, since I've never had an Apple device, so these I've seen recommended elsewhere. Anyway, here are some options:
- If you have an Andorid phone, you can use the Google Drive app: Scan documents with Google Drive. (Android only.)
- I've also used Tinny Scanner, in fact, the Pro (paid) version, and it works well. (Android only.)
- There is also the open source Open Note Scanner, available from F-Droid and Play Store, but I've never used. (Android only.)
- CamSacanner. (Android and iOS.)
- Genius Scan. (Android and iOS.)
- Adobe Scan. (Android and iOS.)
Again, please make sure you are familiar with your scanning app (or real scanner) before the next exam!
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Legal Issues
Conduct
All students should be familiar with Hilltopics' Students Code of Conduct and maintain their Academic Integrity: from Hilltopics Academics:
Academic Integrity
Study, preparation, and presentation should involve at all times the student’s own work, unless it has been clearly specified that work is to be a team effort. Academic honesty requires that the student present their own work in all academic projects, including tests, papers, homework, and class presentation. When incorporating the work of other scholars and writers into a project, the student must accurately cite the source of that work. For additional information see the applicable catalog or the UT Libraries site. See also Honor Statement (below).
All students should follow the Honor Statement (also from Hilltopics Academics):
Honor Statement
"An essential feature of the University of Tennessee, Knoxville, 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 Student 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.
Campus Syllabus
Please, see also the Campus Syllabus.
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Additional Bibliography
Here are some other books you might find helpful:
- J. Fraleigh "A First Course in Abstract Algebra", 7th Ed., 2002. Addison Wesley.
- J. Gallian, "Contemporary Abstract Algebra", 7th Ed., 2009. Brooks Cole.
- M. Artin. "Algebra", 2nd Ed.,2011. Pearson.
- I. Herstein, "Topics in Algebra", 2nd Ed., 1975. Wiley.
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
This is not necessary to our class! I leave it here in case someone wants to learn how type math, for instance to type their HW. But again, you can ignore this section if you want to.
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 freely available for all platforms.
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):
- Cocalc (Previously known as "Sage Math Cloud". This one is much more than just LaTeX.)
- ShareLaTeX
- Overleaf
The first one, Cocalc, is more than just for LaTeX, as you can also run Sage, which can do computations with the objects we will study in this course.
If you want to install LaTeX in your computer (so that you don't need an Internet connection), check here.
A few resources:
- Here is a video I've made where I talk about LaTeX and producing documents with it: Introduction to LaTeX and Sage Math Cloud. (Again, note that "Sage Math Cloud" is simply the old name for Cocalc. The video does not show it in great detail, but might be enough to get you started.) Note it was done for a different course, so disregard any information not about LaTeX itself.
- TUG's Getting Started: some resources, from installation to first uses.
- A LaTeX Primer by D. R. Wilkins: a nice introduction. Here is a PDF version.
- Art of Problem Solving LaTeX resources. A very nice and simple introduction! (Navigate with the links under "LaTeX" bar on top.)
- LaTeX Symbol Lookup: Draw a symbol and the app will try to identify it and give you its LaTeX code.
- LaTeX Wikibook: A lot of information.
- LaTeX Cheat Sheet.
- Cheat Sheet for Math.
- List of LaTeX symbols.
- Comprehensive List of Math Symbols.
- Constructions: a very nice resource for more sophisticated math expressions.
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Links
- Zoom Meeting Links:
- Class: See the Syllabus on Canvas.
- Office Hours: See the Syllabus on Canvas.
- My web pages for Math 351 from Spring 2018, Fall 2017, Spring 2016, Spring 2015 and Fall 2009. Includes all midterms and final with solutions.
- My web pages for Math 506, a course similar to this one, from Summer 2015, Summer 2017, and Summer 2019.
- Canvas.
- Piazza (Math Related Forum).
- Cocalc
- UT Knoxville Home
- UTK's Math Department.
- Services for Current Students and MyUTK (registration, view your grades, etc.).
- Office of the Registrar
- Academic Calendars, including dates for add and drops, other deadlines, final exam dates, etc.
- Hilltopics.
- Students Disability Services
- Office of Equity and Diversity (includes sexual harassment and discrimination).
- My homepage
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Videos
The videos below were made for a different course! So, if you watch them, you have to be careful with comments that I make about the course structure and what is important. They were made for Math 506 -- Algebra for Teachers. That course is taught online and its audience is teachers. Although we cover a lot of the same material, proofs are de-emphasized, unlike this course, where proofs are quite important!
On the other hand, I go over examples and solve problems, so it might be useful for you too. I also go over some computer programs, namely Sage and LaTeX, which are not part of our course, but you can learn them from the videos if you are interested.
If you are uncertain if something from the video is relevant or applies to our course, please ask! (Use a Piazza Forum, please!)
Please let me know if you find any mistake in the videos!
- Introduction:
- Introduction to SageMathCloud: here I show a little about LaTeX and the use of Cocalc.
- Optional: You can watch these videos on What is Algebra?, where I give a brief answer to the question, and on Algebraic Structures (made for Math 457), which goes over a few topics we will cover, but many others that we won't. I think these make a good introduction to our course.
- Section 1.3:
- Long division with negatives.
- Proof of the following Basic Lemma: Suppose that $d \mid a$. Then $d \mid (a+b)$ iff $d \mid b$.
- Euclid's Lemma.
- Example of the Extended Euclidean Algorithm.
- \(b\)-adic representation (1.53 with \(b=4\)).
- Problems: 1.46(ix), (x), 1.60.
- A few words about Example 1.49.
- Computations in Sage.
- Section 1.4:
- Section 1.5:
- General remarks on congruences.
- Powers in congruences.
- Corollary 1.65 (divisibility criteria for 3 and 9).
- Problems 1.77(vi), (vii), 1.78(ii) (and extra example), 1.91(i).
- CRT with non-relatively prime moduli.
- Computations with Sage.
- Section 3.1:
- General Remarks on rings.
- Examples of rings.
- Other operations (subtraction, multiplication by integers, powers).
- Integers modulo \(m\).
- Integral domains (and zero divisors).
- Subrings (including Gaussian integers, \(\mathbb{Q}[\sqrt{2}]\)).
- Units.
- Problem 3.2.
- Computations with Rings in Sage.
- Section 3.2:
- Fraction field.
- Problems 3.17(iv), (vii), 3.19, 3.27(i).
- Section 3.3:
- Watch this video about Section 3.3 (before reading the section!).
- Word about the derivative.
- Section 3.5:
- Section 3.7:
- Section 2.2:
- Section 2.3:
- Section 2.4:
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Handouts
- Campus Syllabus.
- Here is Section 1.3 from our textbook, in case you are still waiting for a copy from the bookstore.
- Here is the Midterm 1 and its solutions.
- Here is the Midterm 2 and its solutions.
- Here is the Midterm 3 and its solutions.
- Here is the Midterm 4 and its solutions.
- Here is the Final and its solutions.
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Homework Problems
Note: This list is subject to change.
Review: Read Sections 1.1 and 1.2 and review things you've forgotten from Math 300/307. From 1.2 you can skip complex numbers if you know the basics: sum, products, absolute value, inverses and geometric representation. We will also use matrices as examples in this course, so maybe a quick review of 251 might be a good idea, especially matrix operations (sums, products, etc.), determinants and inverses.
Section 1.3: 1.46, 1.47, 1.50, 1.53, 1.55(i), 1.57, 1.58, 1.60, 1.62.
Section 1.4: 1.68, 1.69(i), 1.70(i), 1.71, 1.75, 1.76(ii).
Section 1.5: 1.77, 1.78(ii), (iii), (iv), 1.79, 1.81, 1.82(i), 1.85, 1.86, 1.87, 1.88, 1.91, 1.95.
Section 3.1: 3.1 except (v) and (viii), 3.3(i) and (iii), 3.5, 3.6, 3.12, 3.13.
Section 3.2: 3.17 except (vii), 3.19, 3.20, 3.26, 3.27.
Section 3.3: 3.29 except (i), 3.30, 3.32, 3.35 (if you are not familiar with complex numbers, replace them with real numbers, i.e., take alpha to be real), 3.37 (you can use 3.36 without proving it -- also, this one is much easier with the tools from Section 3.5).
Section 3.5: 3.56(i)-(vii) (in (vii) it should say $k=\mathbb{F}_p$, not $k= \mathbb{F}_p(x)$), 3.58, 3.62, 3.64, 3.67 (here $R$ must be a domain, as we usually don't talk about GCDs if the ring of coefficients is not a domain or field).
Section 3.7: 3.86 except (i), 3.87 ((viii) is hard), 3.90(i), 3.91.
Review: Read section 2.1. You should know what it means for a function to be one-to-one (or injective) and onto (or surjective).
Section 2.2: 2.21 ((ii) is easier after Section 2.3), 2.22, 2.23, 2.25, 2.26, 2.27, 2.34 and this extra question.
Section 2.3: 2.36 (i) to (v) and (viii) to (ix), 2.37, 2.38, 2.40, 2.42, 2.44.
Section 2.4: 2.52, 2.54, 2.55, 2.56, 2.57.
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