Math 351: Algebra I - Spring 2018

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Homework Problems
Canvas: important announcements, grades, calendar, etc.
Piazza (Discussion Boards).
Instructor Contact and General Info:
Course Description and Information:
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 11:10-12:10, or by appointment. (Subject to change).




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 at Ayres 123.
Exams:	Midterm 1 (Section 1.3): 02/07 (Wed).
	Midterm 2 (Sections 1.4, 1.5): 02/28 (Wed).
	Midterm 3 (Sections 3.1, 3.2): 03/28 (Wed).
	Midterm 4 (Sections 3.3, 3.5, 3.6): 04/18 (Wed).
	Final: 05/07 from 8am to 10am.
Grade:	20% for each midterm with lowest dropped + 40% for the final.

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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 can choose from this comprehensive list of definitions and theorems the most important (which doing the HW and following the lectures will help to rank) and put them in a 3" by 5" index card (front and back). I will collect those and will allow you to use them (your own) in exams. (I will give them to you before the exams.) I will not check to see if there are any mistakes (thus, be extremely careful when writing, so that you don't have wrong statements when using them on exams!) and they are not worth any points, but they can help you during 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!

Note that these index cards should not be the same as the sheets of paper (with a more comprehensive list), as you will turn them to me, and I will only give them back to you on exams. (I will collect them back after the midterms, but you will keep them after the final.) If the index card is all that you do (and it shouldn't be!), at least make a copy (or take a picture) of it so you can use for future HWs or whenever you need a quick refresher.

You will receive announcements from Canvas telling when to turn these cards in. Some notes about these cards:

When you turn in your first index card, you will turn it in inside a resealable small bag (e.g., a sandwich size zip lock bag) with your name on the bag. I will keep all your future index cards inside this same bag.
The announcement will let you know when, how many and which section(s) you are allowed to put in the card. (Most usually it will be a card per section.)
All index cards must be 3" by 5". (You can cut a piece of paper in this size, though.)
You can write on both sides.
The index card must have your name and the section it pertains clearly written on the top.
The cards must be turned in on the due date! Late cards will not be allowed.
The announcement will only come one or two days before it is due.

Finally, if there is interest, we could set up a collaboration to produce a complete collection of definitions and statements (and theorems). I could set up a LaTeX file (which is a program to typset math -- more on LaTeX below) in Cocalc (that allows us to share LaTeX documents, Google Docs style, and much more) that every student can write on, and together you can produce a complete, nicely formatted list (in PDF format!) for all to use. This does require the use of LaTeX, but I would be glad to help with the formatting and produce some templates in the beginning. If interested, we could divide the work among the students (or at least those who can use LaTeX, e.g., those who have taken Math 171).

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/spring2018/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/spring2018/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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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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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:
- Problems 1.68 (iv), (v).
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:
- Problems 2.36(v), 2.40.
- Subgroups (criteria and examples).
- Problem 2.52(x) and (xi), 2.54(i) (modified), 2.57.

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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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