Math 456: Abstract Algebra II - Spring 2007

Math 456: Abstract Algebra II - Spring 2007



Important Notes


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Final Grades:

The total grades for the course were computed. The average for the total grades was 70.32 and the highest score was 82.1.

If you want to see your final or check your average, make an appointment.

I have already submitted the grades to the registrar office. I suppose that you can already check it at Circle Park Online. I also posted them at Blackboard.


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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 2:30pm-3:30pm or by appointment (subject to change!!)


Textbook: M. Artin. ``Algebra'', 1st Edition. Prentice Hall, 1991.

Prerequisite: Math 455.

Class: MWF 10:10-11:00 at Ayres Hall 111. (Section 1.)

Exams: Midterms: 02/19 (Mon) and 03/30 (Fri), in regular classroom and time; Final: 05/02 (Wed) from 10:15am to 12:15pm, also in our regular classroom.

Grade: 20% for HW + 20% for each Midterm + 40% for the Final. Note the weight of the HWs!


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Midterm I

Here is the Midterm I and the Midterm I with solutions.

The exams were graded and will be returned in class on Friday (02/23). (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.) The average (with the take-home part) was 70.5 and the highest grade was 86.

Due to the small size of the class, I will not post the distribution, but grades above 60 would get at least Cs in a curve for this exam.


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Midterm II

Here is the Midterm II and the Midterm II with solutions [UPDATED: small correction on 3(b) in the exam and solution on 04/13].

The exams were graded and will be returned in class on Wednesday (04/04). (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.) The average (with the take-home part) was 70.75 and the highest grade was 89.

Due to the small size of the class, I will not post the distribution, but grades above 60 would get at least Cs in a curve for this exam.


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Here is the final: Final (take home), Final Solution. The average was 68.25 and the highest grade was 79. (You can check your grades in Blackboard.)

Note that the finals will not be returned, since they have to be kept on file. If you want to see your final or check your average, make an appointment.

There is no point in giving a curve for this grades, since I will curve your averages directly. (See Final Grades above.)


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Course Information:


In this course you will be introduced to Rings, and Fields. I will assume you have the proper background from Math 455. We will use elements from it often. If you had difficulty in that course, you might need to review the material. (I would gladly try to help you with that if you come to my office hours.)


Homeworks will be assigned after every class and will be posted at this page. (See Homeworks below.) No paper copy of the HW assignments will be distributed in class. It is your responsibility to check this page often!

The site above is the official site for the course. Blackboard will be used only for grades. (You will be able to see all your grades there, including HWs.) Everything else will be at this site.

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 at the Homework section below. 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. Please check the Important Notes section soon.

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 Online 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.

Course Structure (UPDATED: 04/24)

We will likely cover the following portions from the text:

Chapter 10 gives the basic properties of Rings, while Chapter 11 studies factorization (in rings). Chapter 13 develops the theory of Fields, and Chapter 13 deals with Galois Theory, which is basically the study of field extensions.

Additional Bibliography

Here are some other books you might find helpful:

The first two books are considered ``easier'' books, and although they also have a somewhat different approach, they have most of the topics we will cover and may be of good help if you have difficulty reading Artin's book.

The last one is a ``standard'' text for a first course in abstract algebra, but might have a higher level of difficulty. (It's been used for the honors section of this course.) Nevertheless, it is a very good reference.  

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Legal Issues


All students should be familiar and maintain their ``Academic Integrity'': from
Hilltopics 2006/2007, pg.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 2006/2007, 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, January 10 - Classes begin.

Monday, January 15 - Marin Luther King Jr. Holiday. (No class.)

Friday, January 19 - Last day to add, change grade options, or drop a full semester course without a "W".

Monday, February 19 - Midterm I.

Tuesday, February 20 - Last day to drop a full term course with a "W".

Monday-Friday, March 12-16 - Spring Break. (No class.)

Friday, March 30 - Midterm II.

Tuesday, April 3 - Last day to drop with a WP/WF.

Friday, April 6 - Spring recess. (No class.)

Friday, April 27 - Last Class Day.

Wednesday, May 02 - 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. They might come with no explanation, just the ``answer''. If yours do not match mine, you can try to figure out again. (Also, read the disclaimer below!) You can come to office hours if you want explanations for the answers. Be careful that just because our ``answers'' were the same, it doesn't mean that you solved the problem correctly (it might have been a ``fortunate'' coincidence), and in the exams what matters is the solution itself. I will do my best to post somewhat detailed solutions, though.

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 01/17 (the official day is Wednesday, but I will accept HWs until Friday in class):

Review rings and ring homomorphisms: Sections 10.1 and part of 10.3. (Follow class notes.)

Section 10.3 (pg. 380): 1 (isomorphism = bijective homomorphism), 2 (unit ideal = whole ring), 4, 6, 14, 20.


HW2 - Due on Wednesday 01/24:

Section 10.3: 8(a) (``describe the kernel'' by giving generators for the ideal -- hint: you need two here), 9 (hint: only one generator here), 19 (hint: use the Binomial Theorem; note that p is prime), 22 (hint: remember than in a field, if ab = 0, then either a or b is zero -- see question 3 of the In Class Final from last semester), 29, 30.


HW3 - Due on Friday 02/02:

Section 10.4: 2, 3, 6, 7(a), 8(a). (These will require things that we'll do in class on Monday. If you need more time, let me know and we can talk about it in class.)


HW4 - Due on Friday 02/16:

Section 10.5: 1, 2, 3, 5, 9.

Section 10.6: 2, 3, 4. (For, 3 and 4, maybe you should prove first that in an integral domain, the degree of a product of two polynomials is the sum of their degrees.)


HW5 - Due on Wednesday 02/28:

Section 10.7: 2, 3, 7, 10.

Section 11.1: 1, 3, 5(a), 8(a), (b).


HW6 - Due on Friday 03/09:

Section 11.2: 1, 2(b), 4, 7, 12, 13.


HW7 - Due on Wednesday 03/21:

Section 11.3: 1, 2, 4, 8 (hint: let R = C[x,y,z] and start working on R[w]), 9 (hint: we can do long division in Z[x], as long as the polynomial we are dividing by is monic).


HW8 - Due on Wednesday 03/28 (UPDATED):

Section 11.4: 1, 3, 4, 8. Optional: 10 (hint: copy the idea in the proof of Eisenstein's Criterion).



HW9 - Due on Wednesday 04/18 (NOTE THE DATE CHANGE AGAIN!):

Section 13.1: 1.

Section 13.2: 3.

Section 13.3: 3(b), (f), 5 (these you can do with the material covered on 04/04), 1, 2, 6, 7(a), (c), 8 (hint:: if m and n are relatively prime, what can you say about common multiples of m and n?), 11.

Section 13.5: 4.


HW10 - Due on Wednesday 04/25:

Section 15.1: 2, 3 (wait until Monday to do this one), 6(a), 6(c), 9, 13 (here, do not assume Theorem 14.1.6) 15 (wait until Monday to do this one).


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 Monday, write me an e-mail at, and I'll update it and let you know.


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