Math 552: Modern Algebra II - Spring 2008

Math 552: Modern Algebra II - Spring 2008



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!) (Updated on 01/24.)

Prerequisite: Undergraduate Abstract Algebra and Math 551.

Class: MWF 10:10-11:00 (or 10:05-11:05) at Ayres Hall 309B. (Section 1.)

Exams: Midterm: 03/03 (Mon), from 10:05pm to 11:05am in our regular room. Final: 05/06 (Tues) from 10:15am to 12:15pm, in our regular classroom.

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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Course Information:

This is the second course of the graduate sequence in Modern Algebra. We will cover topics in Modules and Field/Galois Theory this 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.

As with the last semester, I'd like to propose evening Midterm (at a time convenient to all) and Extended Class Time (start 5 minutes early and finish 5 minutes late). Both of these will be, again, left to the students choice.

Course Structure

We will likely cover parts three and four of the textbook, and revisit a few sections of part five that deals with modules (e.g., localization). Some sections might be left out depending on time (e.g., maybe 11.5 or 13.3), but we will certainly try to cover all topics required for the prelim (as time permits).


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 on the course page. 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.

Unfortunately, I will not post solutions this semester. If you want to see the solution to a problem, please come to office hours.

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 (same I used last semester) 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.


We still have the 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. (Same as last quarter.)

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.''


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, January 09 - Classes begin.

Friday, January 18 - Last day to add, change grade options, or drop a full semester course without a "W".

Monday, January 21 - Marin Luther King Jr. Holiday. (No class.)

Tuesday, February 19 - Last day to drop a full term course with a "W".

Monday, March 03 - Midterm.

Friday-Thursday, March 14-20 - Spring Break. (No class.)

Friday, March 21 - Spring recess. (No class.)

Tuesday, April 1 - Last day to drop with a WP/WF.

Friday, April 25 - Last Class Day.

Tuesday, May 06 - Final.


More dates.


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HW1 - Due on Wednesday 01/16:

Note: Do not use linear algebra (properties of vector spaces) until we have seen it (chapter 11). You can, however use geometry and basic matrix algebra. (And of course, anything that comes from what we have already seen.)

Section 10.1:. Turn in: 8, 13, 15. Do not turn in: (Most of these are quite quick and easy. At least take a look at them.) 2, 3, 4, 5, 7, 18, 19, 20, 21, 23.


HW2 - Due on Wednesday 01/23:

Section 10.2: Turn in: 13. Do not turn in: 4, 6, 8, 9, 10, 11, 12.

Section 10.3: Turn in: 10, 24(c), (e) Do not turn in: 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, 11, 13, 14, 16, 17, 18, 22, 23, 24(a), (b), (d).


HW3 - Due on Wednesday 01/30:

Section 10.4: Turn in: 4, 7.


HW4 - Due on Wednesday 02/06:

Section 10.4: Do not turn in: 3, 6, 9 (use 8(c) without proving it), 10, 11, 15, 17, 18, 24, 25. Turn in: 5, 20 (this one might be tricky; let me know if you need a hint).

Section 10.5: Do not turn in: 1, 10, 11, 12. Turn in: 2.


HW5 - Due on Wednesday 02/13:

Section 10.5: Do not turn in: 4, 6, 7, 8, 9, 13, 14(b). Turn in: 3, 14(a).


HW6 - Due on Wednesday 02/20:

Section 11.1: Do not turn in: 1, 4, 9, 10, 11, 13 (use 12). Turn in: 6.

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

Section 11.3: Do not turn in: 1, 3. Turn in: 4.

Section 11.4: Do not turn in: 1, 2, 3, 6.


HW7 - Due on Wednesday 02/27 (since I've posted these late, you can turn them on Friday if you need more time):

Section 12.1: Do not turn in: 2, 3, 4, 6, 8, 9, 21. (Exercises 16 to 19 are important to justify the algorithm for rational canonical form.) Turn in: 15, 22.

Section 12.2: Do not turn in: 1, 2, 3, 6, 7, 9, 10, 13, 17, 19, 20, 21. (Exercises 22 to 25 are important to justify the algorithm for rational canonical form.) Turn in: 4, 18.


HW8 - Not to turn in (due to the exam):

Section 12.3: Do not turn in: 2, 10, 17, 19, 22, 25, 26, 29, 33. Also make sure you do a few computational ones.

Section 13.1: Do not turn in: 2, 4, 8.

Section 13.2: Do not turn in: 1, 4, 8, 13, 15, 17, 18, 19, 20, 22.


HW9 - Due on Wednesday 03/12:

Section 13.2: Turn in: 17, 22.

Section 13.4: Do not turn in: 1, 2, 3. Turn in: 4.


HW10 - Due on Wednesday 03/26:

Section 13.5: Do not turn in: 2, 3, 6, 8, 9, 10. Turn in: 4, 5.


HW11 - Due on Friday 04/04:

Section 14.9: Do not turn in: 1, 2. Turn in: 3.


HW12 - Due on Friday 04/11:

Section 14.3: (You don't need Galois Theory here, but you can use it if you want.) Do not turn in: 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), 7. Turn in: 6, 10, 11. (Updated!)

Section 14.1: Do not turn in: 1, 2, 3, 4, 5, 6, 7.


HW13 - Due on Friday 04/18:

Section 14.2:Do not turn in: 1, 3, 4, 6 (look at the computations on pg. 557), 7, 8, 9, 13, 14, 15, 16. Turn in: 11.

Section 14.4:Do not turn in: 1, 2, 3, 5(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). Turn in: 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).


HW14 - NOT TO TURN IN (I will suggest some for you to practice, but it won't be turned in):

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

Section 14.5: 3, 5, 6, 7, 8, 9, 10, 11. (Number 12 is also nice, but will not be in the exam.)

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

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

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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