Math 551: Modern Algebra I - Fall 2007

Math 551: Modern Algebra I - Fall 2007



Important Notes


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Instructor Contact and General Information


Instructor: Luís Finotti

Office: Ayres Hall 212-D

Phone: 974-1321 (please do not ask me to call back -- leave your e-mail)


Office Hours: MWF 11:15am-12:15pm or by appointment (subject to change!!)


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

Prerequisite: Undergraduate Abstract Algebra.

Class: MWF 10:05-11:05 at Ayres Hall 309A. (Section 1.)

Exams: 10/22 (Th), from 6pm to 8pm (or 6:30pm to 8:30pm) in room TBA. Final: 12/13 (Thu) from 10:15am to 12:15pm, in our regular classroom. (This will only change if the students decide they want more time.)

Grade: Roughly: 30% for HW + 30% for the Midterm + 40% for the Final. Note the weight of the HWs! (Since this is a graduate course, there will be more leeway on these weights.)


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Here is the Midterm and the Midterm with solutions.

The exams were graded and will be returned in class on Wednesday (10/24). (You can check your grades in Blackboard.) If you are not in class that day, you have to come by my office to pick up yours. (I will not take them to class anymore.)

Since there was (probably) not enough time to work all 4 problems in class, I would consider the in-class part of the exam out of 75. I would be happy with a solid 50, though (i.e., two problems really well-done).

To get your take-home grade, I just replaced the grade of each problem that you've turned in.

Your ``official'' midterm grade was computed by (200*IC/75 + TH)/3, i.e., the in-class grade (out of 75) had twice the weight of the in take-home part (out of 100). I've done this because your grade in this course should reflect how confident I am that you'd pass in your prelim. Moreover, the exam was a bit on the easy side for a take-home.

On the other hand, I was glad to see that many of you obtained very good grades after the take-home. I, myself, don't care that much about how long it takes you, as long as you can do it. But you do have to be prepared for the prelim! If you are among those who did not do as well in class, you should try, every once in a while, to practice working on problems with limited time. (Of course, you will practice more in our future exams.)

The averages were: IC: 46 (out of 75), TH: 81, Official: 68. But again, I'd like to remind you that this grade is not important. What is important is that you prepare yourself for the prelim!


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Course Information:

This is the first course of the graduate sequence in Modern Algebra. We will likely cover topics in Group Theory and Ring Theory in this course, leaving Modules and Field/Galois Theory for the second semester.

The amount to be covered is very large, and thus the pace of the class might be a bit fast. In order to not go too fast, I will assume some background in Groups and Rings. These will be only basic topics that anyone should have seen in an undergraduate algebra course. I might quickly remind you of some of these basic facts, but I might skip some altogether. In this category are included: definitions and basic examples of groups and rings, subgroups and subrings, definitions of normal subgroup and ideal, homomorphisms and isomorphisms, and others. On the other hand, if I skip a fact or definition with which you are not familiar, the students should feel free to stop me and ask about it.

Also, in the same spirit of keeping the pace reasonable, I would like to propose:

Both of these will be left to the students choice, but I should advise you that I'd likely have to go quite fast if either one of these suggestions is not feasible.

Course Structure

We will likely cover the first two parts of the textbook (leaving the latter two for the second semester). These include the first nine chapters. Some sections might be left out depending on time, some we will go over very quickly, since they should be review, but we will certainly cover all topics required for the prelim.


Homeworks will be assigned after every class and will be posted 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!. Besides HW assignments, other important information will be posted here. (Check the section Important Notes often!)

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.

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 this page.

In my opinion, doing the HW is one of the most important parts of the learning process, so the weight for them is equal to the weight of a single midterm, and I will assume that you will 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.


You will have to check your e-mail at least once a week, preferably daily. I will use your e-mail (given to me by the registrar's office) to make announcements. If that is not your preferred address, write me an e-mail letting me know ASAP. I will assume that any message that I sent via e-mail will be read in a week or less, and it will be considered an official communication.


I have an On-line Feedback Form where you can anonymously send me your comments and suggestions. I will consider your comments and try to do whatever I can to resolve possible problems before it is too late. So, please, feel free to use it whenever you have any constructive comment or suggestion. (In fact, I would greatly appreciate it.) If you don't want you comments to be anonymous, just send me an e-mail or come by my office and we can discuss the problem.


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Additional Bibliography

Here are some other books you might find helpful:

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

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.


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Legal Issues

Although this is a graduate course, so I assume we will not have misconduct problems, it is my duty to ask you to read the sections below. (On the other hand, I would consider any kind of lack of academic integrity from a graduate student a much more serious offense, and would feel obligated to take the appropriate measures with maximum rigor.)


All students should be familiar and maintain their Academic Integrity: from
Hilltopics 2007/2008, pgs.39-40:

Academic Integrity

The responsibility for learning is an individual matter. 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 all work presented be the student's own work, not only on tests, but in themes, papers, homework, and class presentation. There is a clear distinction between learning new ideas and presenting them as facts or as answers, and presenting them as one's own ideas. It is part of the learning process to incorporate the thoughts or ideas of others into one's own mind and presentations with the purpose of learning and enlarging on personal boundaries of knowledge.

All students should follow the Honor Statement: from Hilltopics 2007/2008, pg. 11:

Honor Statement

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

You should also be familiar with the Classroom Behavior Expectations.


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

Sexual Harassment and Discrimination

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

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Important dates:


Wednesday, August 22 - Classes begin.

Friday, August 31 - Last day to add, change grade options, or drop a full semester course without a "W".

Monday, September 03 - Labor Day Holiday. (No class.)

Tuesday, October 02 - Last day to drop a full term course with a "W".

Thursday-Friday, October 11-12 - Fall Break. (No class.)

Thursday, October 18 - Midterm.

Tuesday, November 13 - Last day to drop with a WP/WF.

Thursday-Friday, November 22-23 - Thanksgiving. (No class.)

Tuesday, December 04 - Last Class Day.

Thursday, December 13 - Final.


More dates.


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Solutions to Selected HW Problems

Please read: I will try to post here a few solutions. The new solutions will be added to this same file.

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

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


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HW1 - Due on Wednesday 08/29:

Read sections 1.1 to 1.6. (This should be a review.) You don't have to turn in any problems from those sections, but I recommend you try a few to make sure you still remember that material.

Section 1.7: 8, 11, 14, 15, 18, 19, 21, 23.

Section 2.1: 4, 5 (you cannot use Lagrange's Theorem), 14.

Section 2.2: 1, 2, (take a look at 4: you should be able to do it but you do not need to turn it in), 7, 12(b), (d), (e).


HW2 - Due on Wednesday 09/05:

Section 2.3: 1, 8, 9, 11, 12, 16, 23 (use 22 without proof).

Section 2.4: 2, 3, 7, 10, 14(c), (d), 15, 19.

Section 3.1: 1, 3, 9, 14, 21, 28 (use 27 without proof), 29, 41.


HW3 - Due on Wednesday 09/12:

Section 3.2: 4, 10, 11, 14, 18, 19, 21.

Section 3.3: 1, 3, 7, 10.


HW4 - Due on Wednesday 09/19:

Section 3.4: Turn in: 4, 5, 10. Do not turn in: 2, 7.

Section 3.5: Turn in: 12. Do not turn in: 3, 4, 9.

Section 4.1: Turn in:7(d), 9. Do not turn in: 1, 7(a)-(c).

Section 4.2: Turn in: 11. Do not turn in: 12, 13.

Section 4.3: Postponed!


HW5 - Due on Wednesday 09/26:

Section 4.3: Turn in: 26, 30. Do not turn in: 1, 2, 27, 29.

Section 4.4: Turn in: 8(b), (c), 14. Do not turn in: 13, 18, 20 (you can skip the QD_16 group).

Section 4.5: Turn in: 12, 33, 36. Do not turn in: 2, 32.


HW6 - Due on Wednesday 10/03:

Section 4.5: Turn in: 17, 25, 44. Do not turn in: 15, 16, 22, 27.

Section 4.6: Turn in: 2, 3. Do not turn in: 1, 4.

Section 5.1: Just browse to see the problems.

Section 5.2: Turn in: 5. Do not turn in: 2, 3, 6, 9.

Section 5.4: Turn in: 5. Do not turn in: 4, 11.


HW7 - Due on Wednesday 10/10:

Section 5.5: Turn in: 6, 8, 20, 24. Do not turn in: 1, 2, 18, 21.


HW8 - Due on Wednesday 10/17:

Section 6.1: Turn in: 4, 7, 9, 21. Do not turn in: 3, 10, 12, 25.


HW9 - Not to be turned in (since you have the exam, but work on them anyway!):

Section 6.3: Do not turn in: 1, 3, 11.

Section 7.1: Do not turn in: 3, 8, 13, 14, 15, 23, 25(c). Also, take a look at others. Many should be quite easy.


HW10 - Due on Wednesday 10/31:

Section 7.2: Do not turn in: 3, 4, 5, 10.

Section 7.3: Turn in: 13, 17, 29, 34(d). Do not turn in: 3, 5, 8, 22, 24, 26, 34(c).

Section 7.4: Turn in: 13, 25. Do not turn in: 2, 10, 11, 15, 16, 19.


HW11 - Due on Wednesday 11/07:

Section 7.4: Turn in: 36, 39, 40. Do not turn in: 38, 41.

Section 7.5: Turn in: 5, 6. Do not turn in: 4.

Section 15.4: Turn in: 22. Do not turn in: 21, 23.


HW12 - Due on Wednesday 11/14:

Section 7.6: Turn in: 1,7. Do not turn in: 3. Also, take a look at 5(b) and (c).

Section 8.1: Turn in: 3, 10. Do not turn in: 4, 8(a), 11. Take a look at 1 and 2.

Section 8.2: Turn in: 3, 8. Do not turn in: 6.


HW13 - Due on Wednesday 11/21:

Section 8.3: Turn in: 5(b),(c). Do not turn in: 5(a), 9, 11.

Section 9.1: Turn in: 11. (It was suppose to be 17...) Do not turn in: 8. Take at look at some of the easy/compuational ones.

Section 9.2: Turn in: 12(b). Do not turn in: 4, 12(a), (c). Take at look at some of the easy/compuational ones.

Section 9.3: Turn in: 4(b). Do not turn in: 1, 2, 4(a), (c), (d).

Section 9.4: Turn in: 3. Do not turn in: 4, 12. (12 is the only one that requires things I haven't finished. So, you can leave that for Monday.) Also look at the easy/computational ones, such as 1, 2, 9, 10, 11, 18. (Many of these need the Eisenestein Criterion, so you may leave them for Monday too.)


HW14 - Due on Wednesday 11/28:

Section 9.5: Turn in: 7. Take at look at the others.

Section 9.6: Do not urn in: 1.

Section 15.1: Turn in: 1. Do not turn in: 2, 4. (You might need the Hilbert basis for these, so leave them for Monday. But you don't need it for 1.)


HW15 - Not due! Just work on them to prepare for the final:

Section 16.1: Do not turn in: 2, 3, 4.

Section 16.2: Do not turn in: 1, 3, 4.

And that's all!


PLEASE, HIT ``REFRESH'' (OR ``RELOAD'') IN YOUR BROWSER WHEN VISITING THIS PAGE!!!!!!! I usually get messages asking for the update in the HW when it has already been updated. Since I change this page often, some times the browser don't see the changes. But, if you hit refresh and there is still problems missing, feel free to write me.


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 at, and I'll update it and let you know.


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