## Navigation

- Canvas: important announcements, grades, calendar, etc.
- Ed (Discussion Boards).
**Homework Problems**- Instructor Contact and General Info:
- Course Information
- Course Policies
- Legal Issues
- LaTeX
- Videos
- Links
- Handouts
- Homework Problems

## 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: |
By appointment only. Mask required for in-person. Otherwise, we can meet with Zoom. |

Textbook: |
J. Rotman, "A First Course In Abstract Algebra", 3rd Edition, Prentice Hall, 2006. |

Prerequisites: |
Math 300/307 (and 251/257). |

Class Meeting Time: |
MWF 9:15-10:05 at Ayres 123. |

Exams: |
Midterms: 02/16, 03/09, 04/06, 04/27 Final: 05/16, 10:30am. |

Grade: |
10% for HW, 20% for each Midterm dropping lowest, 30% 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**: Basic Number Theory*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**: Rings, Fields, Polynomial Rings*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**: Groups*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*!

### Additional Bibliography

Here are some other books, besides out text, that 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.
- T. W. Judson "Abstrac Algebra",
*freely available*! - M. Artin, "Algebra", 2nd Ed.,2011. Pearson.
- I. Herstein, "Topics in Algebra", 2nd Ed., 1975. Wiley.

The first three 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 has a higher level of difficulty than the previous three. It’s been used for the honors section of the undergraduate algebra course here at UT.

## Course Policies

### Masks

I would like to *ask* you to wear masks in all our classes, even if not
required by the university. **If not required by UT, this will not be
enforced in any way and there will be no penalty for not wearing
masks.**

But keep in mind that others in the class might be, or have people close to them, that have higher risk of being affected by COVID, such as the elderly, people with compromised immune systems, or people that due to allergies cannot take the vaccine. Even if you don’t believe that masks make a difference (even if they really do, see: CDC, Mayo Clinic, Johns Hopkins Medicine to just cite a few), I’d ask you to still do it as a sign of respect and consideration.

What I *can* (and will) do is to require masks for *in-person* office
hours. If you are not willing to wear a mask, **I still can help you
over Zoom.** Just write me an email and make an appointment.

### Homework Policy

Homework will be assigned weekly and it will usually be due on Wednesdays. The official due dates will be posted in Canvas (as an Assignment). The problems will be those from the Homework Problems section below.

Here is how it will work: the sections we cover in a week are due in
the next Wednesday. For example, if in week 4 we cover Sections 3.1
and 3.2, the problems for those sections in Homework
Problems will be due in the Wednesday of week 5.
Although I will post the corresponding due dates on Canvas, these will
only come on Fridays, when I know exactly what was covered.
Therefore, **do not wait for me to post the due dates**! As soon as I
finish a section, start working on the problems from that section. If
you leave it all to start Friday (after I post the assignment), you
won’t have as much time to do all the problems.

To avoid cheating/copying, **HW sets have to be handwritten** and
**not photocopied**. You do *not* need to copy the statement of
the problems: just make clear the section and problem number.

**HW problems will be basically graded for completion!** I will only
spot check most sets for major problems. You get 2 points for a
properly complete HW set, 1 for somewhat incomplete or sets having problems,
and 0 for not turned in, mostly empty, or highly problematic sets.

Since these sets are not properly graded, solutions will be posted in Canvas.

To help you take your HW seriously, **there will be at least one
problem straight from the HW in each exam**. In my opinion, *doing the
HW is the most important part of the course*, as you can only learn by
solving problems, so I urge you to take it seriously.

### Exams

We will have four midterms and a final. The lowest score of the midterms will be dropped when computing your course average.

Here are the exams (sections are **tentative!**):

**Midterm 1:**On 02/16, covering Section 1.3.**Midterm 2:**On 03/09, covering Sections 1.4 and 1.5.**Midterm 3:**On 04/06, covering Sections 3.1, 3.2, and 3.3.**Midterm 4:**On 04/27, covering Sections 3.5, 3.6, and 3.7.**Final:**On 05/16 at 10:30am, covering all sections (comprehensive).

Each exam will contain *at least* one question coming straight from a
HW set.

You will also be allowed to bring index cards to our exams. (See below.)

### 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.

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. It can be typed, or photocopied, if you prefer.

*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!

### Curve

In principle there will be no curve for this course. The letter grade cut offs can be found here. (Basically 90 and above is an A, 80 and above is a B, etc.)

In *special circumstances* I might curve the final course averages
before submitting the letter grades, but this *should not be
expected*. On the other hand, a curve can only help with the grades.

Even if I do decide to curve the course averages, I will not curve midterm grades, as since we drop the lowest midterm grade, a curve for a midterm would not help you predict your course grade.

### 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 3.5, 3.6, and 3.7, and say that in our final questions 3 and 4 (and only those) are about the material of those chapters. 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, at the Absence Justifications
Page. 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.)

Late HW sets will only be accepted with proper justification (subject to approval) and each student can turn in a late HW at most once in the semester.

### How to Be Successful in this Course

*Study hard and do as many problems as you can!*- 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 Ed 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.

A couple more things to keep in mind:

- Studying a little every days is better than cramming before the exam.
- Reviewing your class notes shortly after class will help you retain what you’ve learn.
- It’s a lot easier to remember procedures and statements of theorems if they make sense. So try to understand what we are doing instead of just memorizing it.

### Ed (Discussion Board)

We will use Ed for online discussions. The advantage of Ed (over other discussion boards) is that it allows us (or simply me) to use math symbols efficiently and with good looking results (unlike Canvas) and to post (and run!) formatted code. It also allows anonymous posts (also unlike Canvas).

To enter math, you can use LaTeX code. (See the section on LaTeX below.) Even if you don’t take advantage of this, I can use it making it easier for you to read the answers.

You can access Ed through here: https://edstem.org/us/courses/18575/discussion/. (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 categories for our discussions:

*Lectures:*Questions about something I’ve done in class.*Homework:*Questions about the HW.*Exams:*Questions about the exams.*Course Structure:*Ask questions about the class, such as “how is the graded computed”, “when is the final”, etc. in this category. (Please read the Syllabus first, though!)*Feedback:*Give (possibly anonymous) feedback about the course using this category.*Computer:*I might post some notes on how to do some computations using a computer. Or you can ask about LaTeX as well.*Other:*In the unlikely event that your question/discussion doesn’t fit in any of the above, please use this category.

I urge you to use Ed 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 Ed. Discussion is encouraged here!

Also, please don’t forget to choose the appropriate category for your
question. And make sure to choose between *Question* and *Post*.

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. If you have a comment, question, or suggestion, you
can use the *Comment* area.

You can also use Ed for *Private Posts*, by checking the corresponding
box. Posts marked as private will be only viewed by the student who
posted and me. Only use this what you have to ask cannot be shared
with all, e.g., if you are sharing something from your HW. Otherwise,
don’t make it private, as other students might have the same questions
as you.

You should receive an invitation to join our class in Ed via your
“@vols.utk.edu” e-mail address before classes start. If you don’t,
you can sign up here: https://edstem.org/us/join/5F4t2y. If
you’ve register with a different e-mail (e.g., @tennessee.edu) you do
not need to register again, but you can consolidate your different
e-mails (like @vols.utk.edu and @tennessee.edu) in Ed, 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 *Account* icon on the top right of Ed, select *Emails*, and
then *Add email address*.

### Communications and E-Mail Policy

You are *required* to set up notifications for Ed and for Canvas to be sent
to you *immediately*.

On Ed, click on the *Account* icon on the top left, then *Settings*.
In the new page click on *Notifications*. Under *New Thread Digest*,
set the drop down box to *Instant*. *I will consider a post in Ed
official communication in this course, I will assume all have read
every single post there!*

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 Ed, 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
Ed. Usually these should be posted as *Post* and put
in the *Feedback* category. These can be posted anonymously (or not),
*just make sure to check the appropriate option*. **Other 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 Ed (possibly anonymous).

## Legal Issues

### Conduct

All students should be familiar with
Hilltopics’ Students Code of
Conduct and
maintain their *Academic Integrity*: from
Hilltopics
Academics:

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 the Student Code of Conduct and 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.

### Discrimination and Harassment

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

### Campus Syllabus

Please, see also the Campus Syllabus.

## 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, and will be used in our course.)
- 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.

## 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 Ed,
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.

- Introduction to SageMathCloud:
here I show
**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.

- Watch this video about Section
3.3 (
**Section 3.5:****Section 3.7:****Section 2.2:****Section 2.3:****Section 2.4:**

## Links

- Canvas.
- Ed (Math Related Forum).
- Here are the sites for the previous times I taught this course. These contain old exams with solutions.
- My web pages for Math 506, a course similar to this one. These contain old exams with solutions.
- 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

## Handouts

- Campus Syllabus.
- Here is Midterm 1 and its solutions.
- Here is Midterm 2 and its solutions.
- Here is Midterm 3 and its solutions.
- Here is Midterm 4 and its solutions.

## Homework Problems

Due dates and solutions will be posted in Canvas.

**Note:** Although unlikely, 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 (for (vii), note that $\mathbb{Q}[\sqrt{2}] = \{ a + b \sqrt{2} \; : \; a, b \in \mathbb{Q}\}$ – see also problem 3.22), 3.19, 3.20, 3.22, 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.