## Navigation

**Homework**- Latest Homework Assigned
- Canvas: important announcements, solutions, grades, calendar, etc.
- Piazza (Discussion Board).
- Instructor Contact and General Info:
- Course Description and Information:
- Legal Issues:
- Additional Bibliography
- LaTeX
- Links
- Problems Likely To Be Assigned
- 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 from 11:10 to 12:10, or by appointment. |

Textbook: |
D. Dummit and R. Foote, Abstract Algebra, 3rd edition, 2003, Wiley. (ERRATA!) |

S. Lang, Algebra, Revised 3rd edition, 2005, Wiley. (A free electronic copy is available online from the library.) | |

Prerequisite: |
Math 551, or a graduate course in Groups and Rings. |

Class Meeting Time: |
MWF 10:10-11 at Ayres 113. |

Exams: |
Midterm: 03/04 (Monday) during class. |

Final: 05/06 (Monday) from 8am to 10am. | |

Grade: |
20% for HW, 30% for the Midterm, 50% for the Final. |

See here for letter grade ranges. |

Back to the TOP.

## Course Information

### Course Content

This is the second course of the graduate sequence in Abstract Algebra. We will likely cover topics in Modules and Fields, including Galois Theory, in this course, leaving Modules and Fields/Galois Theory for the second semester.

The amount to be covered is again *very* large, and thus the
pace of the class might be a bit fast. **I will assume you still
remember Groups and Rings, and have some familiarity with Vector
Spaces and Fields.** For the latter two, I will only assume
that you know basic topics that anyone should have seen in an
undergraduate algebra course, or mentioned last semester. I might
quickly remind you of some of these basic facts, but I might skip some
altogether. **Please, slow me down if I'm going too
fast.**

### Chapters and Topics

We should cover Chapter 10 (from Dummit and Foote), skipping (perhaps) Injective and Flat Modules from Section 10.5 and Chapter 12 for Modules. Chapter 11 is about Vector Spaces, I will not go over it in detail, but might give you some reminders of the basic properties. (It might be a good idea for you review it yourself, excluding Section 11.5.)

For Field Theory, we will switch to Lang's book. We will cover Chapter V (all sections) and Chapter VI, from Section VI.1 to Section V.7. (If time allows, also Sections VI.8 and VI.9.) This roughly corresponds to Sections 13.1-6 and 14.1-8 from Dummit and Foote. (The HW will still be from Dummit and Foote.)

### Homework Policy

Homework will be posted regularly at the
section Homework of this page. No paper copy of the
HW assignments will be distributed in class.
**It is your responsibility to check this page often!** I will
try to also add the due dates to Canvas's Calendar, but this page has
the official assignment (and due dates).

The HWs will be collected on Wednesdays. Each HW will have
problems from the previous week (Monday, Wednesday and Friday
lectures). The problems to be turned in, as well as due dates, will
be clearly posted here. Note that not all of the problems turned in
will be graded, but you won't know which until you get them back. I
will also recommend extra problems that you do not have to turn in.
On the other hand, *I very strongly recommend that you do those
problems too*, especially if you plan to take the Algebra
Prelim.

Problems likely to be assigned are posted below, and so, although they are subject to change, it is not likely it will happen often. So, you can always start working on the HW early, even if the assignment is not posted.

Note that I might sometimes get too ambitious in posting problems, i.e., I might think we will cover a section during the week, put exercises from it in the next assignment, and then end up not being able to finish it. In this case I might have to take a few problems off the assignment. The bottom line is the following: the assignment is not final until I remove the "More to come" from it. (If you've done problems which were removed, just saved them for the following week.)

Finally, if there is still a "More to come" in an assignment on a Friday, please write me right away so that I can update it. If I delay in replying, you can proceed with the Problems Likely To Be Assigned.

**No late HWs will be accepted**, except in extraordinary
circumstances which are properly documented.

**It is your responsibility to keep all your graded HWs and
Midterms!** It is very important to have them in case there is any
problem with your grade.

I will do my best to post solutions. If I do, they will be posted in Canvas. If I do not and you have a question, you can come talk to me.

In my opinion, doing the HW is one of the most important parts of the learning process, so I will assume that you will (and urge you to) work very hard on them.

Also, you should try to come to my office hours if you are having
difficulties with the course. I will do my best to help you. Please
try to come during my *scheduled* office hours, but feel free to
make an appointment if that would be impossible.

### Piazza (Discussion Board)

We will use Piazza for discussions. The advantage of Piazza is that it allows us (or simply me) to use math symbols efficiently and with good looking results (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 it, making it easier for
you to read the answers.

You can access Piazza through the link on top of this page or directly here: https://piazza.com/utk/spring2019/math552/home. (There is also a link at the Links section.)

To keep things organized, I've set up a few different folders/labels for our discussions:

*HW:*Ask*math*question about HW.*Midterm and Final:*Ask questions about the exams.*Modules and Fields:*Ask question about the theory of modules and fields.*Class 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!)*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 edited 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/spring2019/math552. 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 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.

**Important:** Please do not use Piazza for math questions if
you can come see me in person (especially during office hours).
You will benefit much more if you come see me! A five minute
conversation will be much more productive that a half-hour exchange
in Piazza.

### 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 Account button on
the top left, 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).

Back to the TOP.

## 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 studentâ€™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 alsoHonor 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.

## Additional Bibliography

Here are some other books you might find helpful:

- S.
Lang. "Algebra",
3rd Edition. Springer, 2005. -- Probably the best
*reference*algebra book there is. - I. Isaacs, "Algebra: A Graduate Course", 1st Ed., 1994. AMS (Originally published by Brooks Cole/Cengage Learning). -- Particularly good in group theory and non-commutative algebra.
- B.L. van der Waerden, "Algebra I, II", 2nd Ed., 2003, Springer. -- A classic.
- N. Jacobson, "Basic Algebra I and II", 2nd Ed., 1985. W H Freeman & Co. (Reprint by Dover.) -- Another standard book.
- T. Hungerford, "Algebra", 1st Ed., 1974, Springer. -- Another standard book.

Here are some which are more on the level of *undergraduate*
algebra:

- 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, and it might be even on the level of a graduate course in some parts.

Back to the TOP.

## 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 available for all platforms and freely available.

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

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

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

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

- https://cocalc.com/ (This one is much more than just LaTeX. We will use this one for our meetings.)
- https://www.sharelatex.com/
- https://www.overleaf.com

We will use the first
one, CoCalc in our
course, so *you have to register for it*, and thus might as
well use it. It is probably the best of the services anyway, and it
can do a lot more than just LaTeX. You should have received, by the
first day of classes, an invitation to collaborate on a project that
I've created for this course (Math 504 -- Summer 2018).

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

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

A few resources:

- Here is a video I've made for Math 506 where I talk about LaTeX and producing documents with it: Introduction to LaTeX and Sage Math Cloud (noting, again, that Sage Math Cloud was the previous name for CoCalc). (Not in great detail, but might be enough to get you started.) Note it was done for Math 506 from Summer 2015, so a few things are different. Particularly, we will use "Class_Discussions.tex" instead of "Questions.tex" for our online meetings.
- 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.

Back to the TOP.

## Links

- Math 551 (Fall 2018).
- My 551/552 courses from 2014/2015: M551 (Fall 2014) and M552 (Spring 2015) course pages.
- My 551/552 courses from 2007/2008: M551 (Fall 2007) and M552 (Spring 2008) course pages.
- My undergraduate
*honors*algebra course pages from 2013/2014: M457 (Fall 2013) and M458 (Spring 2014) course pages. - My undergraduate algebra course pages from 2006/2007: M455 (Fall 2006) and M456 (Spring 2007) course pages.
- My undergraduate algebra course pages for M455 (Fall 2012).
- Canvas.
- Piazza
- 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

Back to the TOP.

## Handouts

- Campus Syllabus.
- Field/Galois Theory Statements
- Example of finite extension with infinitely many intermediate extensions.
- Proposition about the minimal polynomial stated in class.
- Radical Extensions Are Solvable.
- Defining Automorphisms.

Back to the TOP.

## Problems *Likely* To Be Assigned

Here are some review problems from Chapter 11 (Vector
Spaces). **These will not be assigned to be turned in!**

**Section 11.1:**
1, 4, 6, 9, 10, 11, 13 (use 12).

**Section 11.2:**
8, 10, 17, 22, 23, 24, 27, 31, 38, 39. (I did not put computations in here, but you should be able to do them...).

**Section 11.3:**
1, 3, 4.

**Section 11.4:**
1, 2, 3, 6.

This list is subject to change without prior notice. The official assignments will be posted below.

**Section 10.1:**. (Most of these are quite quick and easy. At least take a look at them.) 2, 3, 4, 5, 7, 8, 13, 15, 18, 19, 20, 21, 23.

**Section 10.2:** 4, 6, 8, 9, 10, 11, 12, 13.

**Section 10.3:** Look at all of them, and do a few. There are too many problems that show nice (and easy) properties of modules. 1, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 18, 22, 23, 24(a)-(e).

**Section 10.4:** 3, 4, 5, 6, 7, 9 (use 8(c) without proving it), 10, 11, 15, 17, 18, 20, 24, 25.

**Section 10.5:** 1, 3, 4, 6, 7, 8, 9, 10, 11, 12 13, 14(a)-(b).

**Section 12.1:** 2, 3, 4, 6, 8, 9, 15, 21, 22. (Exercises 16 to 19 are important to justify the algorithm for rational canonical form.)

**Section 12.2:** 1, 2, 3, 4, 6, 7, 9, 10, 13, 17, 18, 19, 20, 21. (Exercises 22 to 25 are important to justify the algorithm for rational canonical form.)

**Section 12.3:** 2, 10, 17, 19, 22, 25, 26, 29, 33. Also make sure you do a few computational ones.

**Section 13.1:** 2, 4, 8.

**Section 13.2:** 1, 4, 8, 13, 15, 17, 18, 19, 20, 22.

**Section 13.4:** 1, 2, 3, 4.

**Section 13.5:** 2, 3, 4, 5, 6, 8, 9, 10.

**Section 14.9:** 1, 2, 3.

**Section 14.3**: (You don't need Galois Theory here, but you can use it if you want.) 3, 4, 5 (this statement is not so good -- identity is an isomorphism -- so try to show that the roots of the second polynomial are in the splitting field of the first), 6, 7, 10, 11.

**Section 14.1**: 1, 2, 3, 4, 5, 6, 7, 9.

**Section 14.2**: 1, 3, 4, 6 (look at the computations on pg. 557), 7, 8, 9, 11, 13, 14, 15, 16..

**Section 14.4**: 1, 2, 3, 4 (the hint suggests that *f* is separable; the statement is true in general, and so do it for the general case; the hint seems to go bad in this situation, so maybe you shouldn't try to follow it; finally, note that for the book, Galois implies *finite*, so assume that *K/F* is finite), 5(a)-(b), 6 (note that this is the hard way; it's easier to show explicitly that it is not simple), 7, 8 (I think it needs a *little* of flat modules).

**Section 13.6**: 1, 2, 3, 5, 6.

**Section 14.5**: 3, 5, 6, 7, 8, 9, 10, 11, 12.

**Section 14.6**: 2(b), (c), 4, 5, 10, 18, 20, 28.

**Section 14.7**: 3, 4, 5, 6, 7, 8, 12, 13.

Back to the TOP.

## Homework

**HW1** - Due on **Wednesday 01/16**:

**Section 10.1:** 8, 13.

**Section 10.2:** 6, 9.

**HW2** - Due on **Wednesday 01/23**:

**Section 10.3:** 13, 14, 17.

**HW3** - Due on **Wednesday 02/06**:

**Section 10.4:** 10, 20, 25.

**HW4** - Due on **Wednesday 02/20**:

**Section 10.5:** 8, 9, 14(a).

**HW5** - Due on **Wednesday 02/27**:

**Section 12.1:** 2, 15, 21

**HW6** - Due on **Wednesday 03/13**:

**Section 13.2:** 15, 17, 18(a).

**HW7** - Due on **Wednesday 04/03**:

**Section 13.5:** 4, 5, 6.

**Section 14.1:** 4, 5, 9.

**HW9** - Due on **Wednesday 04/17**:

**Section 14.2:** **More to come!**

If it is already **Friday** afternoon and there still is a
"More to come" after the HW assignment due on the coming Wednesday,
write me an e-mail to lfinotti@utk.edu, and I'll
update it and let you know.

Back to the TOP.