Math 551: Modern Algebra I - Fall 2014

Math 551: Modern Algebra I - Fall 2014



Instructor 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!)


Office Hours: MW 9-10 or by appointment.


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

Prerequisite: Undergraduate Abstract Algebra.

Class: MWF 10:10-11:00 at Ayres Hall 122. (Section 1.)

Exams: Midterm 10/15 (Wed) in class (+ take home part). Final: 12/08 (Monday) from 8am to 10am, in our regular classroom.

Grade: Roughly: 28% for HW + 27% for the Midterm + 45% 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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Course Information

Course Content

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.

To be more precise, the following sections from our textbook, will be either assumed (and skipped) or only partially covered or reviewed:

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.

Chapters and Topics

I'll assume you are familiar with the material of Chapters 1 to 3, but I will, very briefly talk a little bit about sections 1.2, 1.7, 2.2, 3.3. We should then cover Chapters 4 and 5, section 6.1 (and maybe 6.3) on groups.

On rings, I will assume you are mostly familiar with the content of sections 7.1-3 and a little familiar with section 7.4. I will say a few words about group rings (section 7.3) and section 7.4, though. Then, we should cover sections 7.5-6, Chapters 8 and 9 (except for Grobner basis on 9.6). We should also cover parts of sections 15.1-4 and 16.1-2.


Homework Policy

Homeworks 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!.

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!

Problems likely to be assigned are posted below, but are subject to change. So, you can always start early, even if the assignment is not posted. (Chances of changes are small.)

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 this page. 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 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.

E-Mail Policy

I will assume you check your e-mail at least once a day. I will use your e-mail (given to me by the registrar's office) to make announcements. (If that is not your preferred address, please make sure to forward your university e-mail to it!) I will assume that any message that I sent via e-mail will be read in less than twenty four hours, and it will be considered an official communication.

Blackboard and Feedback

There is a link for (general) Feedback on Blackboard. That is in fact a discussion board and I'd ask you to subscribe to it to receive e-mail notifications for posts. Please, post all comments and suggestions there as often as you want. (I really appreciate your input.) These can be posted anonymously (or not). Just make sure to check the option before posting! 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, but then, of course, it won't be anonymous.


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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 in some parts.


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Legal Issues


All students should be familiar and maintain their Academic Integrity: from Hilltopics, pg. 46:

Academic Integrity

The university expects that all academic work will provide an honest reflection of the knowledge and abilities of both students and faculty. Cheating, plagiarism, fabrication of data, providing unauthorized help, and other acts of academic dishonesty are abhorrent to the purposes for which the university exists. In support of its commitment to academic integrity, the university has adopted an Honor Statement.

All students should follow the Honor Statement: from Hilltopics, pg. 16:

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.

We are in a honor system in this course!



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.


Campus Syllabus

Please, see also the Campus Syllabus.


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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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Problems Likely To Be Assigned

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


Section 3.4: 2, 4, 5, 7, 9, 10.

Section 3.5: 3, 4, 9, 12.

Section 4.1: 1, 7(a-d), 9.

Section 4.2: 11, 12, 13.

Section 4.3: 1, 2, 26, 27, 29, 30.

Section 4.4: 8(b-c), 13, 14, 18, 20 (you can skip the QD_16 group).

Section 4.5: 2, 12, 15, 16, 17, 22, 25, 27, 32, 33, 36, 44.

Section 4.6: 1, 2, 3, 4.

Section 5.1: Just browse to see the problems.

Section 5.2: 2, 3, 5, 6, 9.

Section 5.4: 4, 5, 11.

Section 5.5: 1, 2, 6, 8, 18, 20, 21, 24.

Section 6.1: 3, 4, 7, 9, 10, 12, 21, 25.

Section 6.3: 1, 3, 11.

Section 7.1: 3, 8, 13, 14, 15, 23, 25(c). Also, take a look at others. Many should be quite easy.

Section 7.2: 3, 4, 5, 10.

Section 7.3: 3, 5, 8, 13, 17, 22, 24, 26, 29, 34(c-d).

Section 7.4: 2, 10, 11, 13, 15, 16, 19, 25, 36, 38, 39, 40, 41.

Section 7.5: 4, 5, 6.

Section 15.4: 21, 22, 23.

Section 7.6: 1, 3, 5(b-c), 7.

Section 8.1: 1, 2, 3, 4, 8(a), 10, 11.

Section 8.2: 3, 6, 8.

Section 8.3: 5(a-c), 9, 11.

Section 9.1: 8, 17. Take at look at some of the easy/compuational ones.

Section 9.2: 4, 12(a-c). Take at look at some of the easy/compuational ones.

Section 9.3: 1, 2, 4(a-d).

Section 9.4: 3, 4, 12. Also look at the easy/computational ones, such as 1, 2, 9, 10, 11, 18.

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

Section 9.6: 1.

Section 15.1: 1, 2, 4.

Section 16.1: 2, 3, 4.

Section 16.2: 1, 3, 4.


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HW1 - Due on Wednesday 08/27:

Section 1.7: Not to turn in: 8, 11, 14, 15, 18, 19, 21, 23.

(So, there is no HW to turn in on 08/27.)


HW2 - Due on Wednesday 09/03:

Section 3.3: Not to turn in: 1, 3, 7, 10.

(So, there is no HW to turn in on 09/03 either..)


HW3 - Due on Wednesday 09/17:

Section 3.4: Turn in: 4, 7, 9. Do not turn in: 2, 5, 10.


HW4 - Due on Wednesday 09/24:

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

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

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


HW5 - Due on Wednesday 10/01:

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

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


HW6 - Due on Wednesday 10/15:

Section 4.5: Turn in: 22, 32. Do not turn in: 2, 12, 15, 16, 17, 25, 27, 33, 36, 44.

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


HW7 - Due on Wednesday 10/22:

Section 5.1: Do not turn in: Just browse to see the problems.

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

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


HW8 - Due on Wednesday 10/29 (none are to be turned in due to the take-home exam, but work on it and check with the solutions):

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


HW9 - Due on Wednesday 11/12:

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

Section 7.2: Turn in: None. Do not turn in: 3, 4, 5.

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


HW10 - Due on Wednesday 11/19:

Section 7.4: Turn in: 15, 25, 30. Do not turn in: 10, 11, 13, 16, 19, 36, 38, 39, 40, 41.


HW11 - Due on Wednesday 11/26:

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

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

Section 7.6: Turn in: 1. Do not turn in: 3, 5(b-c), 7.


HW12 - NOT TO BE TUNED IN! Just practice for the final:

Section 8.1: 1, 2, 3, 4, 8(a), 10, 11.

Section 8.2: 3, 6, 8.

Section 8.3: 5(a-c), 9, 11.

Section 9.1: 8, 17. Take at look at some of the easy/compuational ones.

Section 9.2: 4, 12(a-c). Take at look at some of the easy/compuational ones.

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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