Math 351: Algebra I - Fall 2009

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## Important Notes

• (12/09 - 10:50am) Your average will be computed in the following way. I will drop your lowest HW/Quiz score when computing your Quiz/HW average. Then, if you did not miss more than a single HW/Quiz, then your course average will be the greatest value between 0.25*HW + 0.20*(Mid1 + Mid2) + 0.35*Final (the way made explicit in the syllabus) and (0.20*(Mid1 + Mid2) + 0.35*Final)/0.75 (so, the HW are not considered). The bottom line is that the HW can only help you. On the other hand if you missed more than one Quiz/HW, your course average will be computed as in the syllabus (i.e., 0.25*HW + 0.20*(Mid1 + Mid2) + 0.35*Final).

If you did not turn in your HW/Quiz for health reasons (and I told you that I would replace your score with your final), then this score will count (as usual), i.e., you will not be penalized.

• (12/01 - 9:35am) I have just written the final exam. Here is some info about it:

• It will be an in-class exam (with no take home part), and it will be on 12/09 (Wednesday), from 10:15am to 12:15pm, in our regular classroom.
• I will allow calculators again, but as before you'd need to justify your answer. On the other hand, you don't need a calculator to be able to solve the problems.
• It covers the whole material of the course, more precisely, Sections 1.3 (up to pg. 48), 1.4, 1.5 (up to pg. 70, excluding examples 1.66 and 1.78), 3.1, 3.2, 3.3 (I will not ask anything about polynomials seen as sequences), 3.5-6 (selected topics -- compare with your notes), 3.7 (expect ``algebraic integers''), 2.2. Of course, you can skip things that involve complex numbers (as we did) and other topics that we've skipped. (Maybe your notes would be a better reference.)
• The exam has 8 questions: one from Chapter 1 (10 points), one from Sections 3.1-2 (multiple parts, worth 20 points total), one from Section 3.3 (10 points), two from Section 3.5 (10 points each), two from Section 3.7 (10 points each), and one from Section 2.2 (with multiple parts, worth 20 points total). There are five computational questions (60 points) and three proofs (40 points).
• The problems are very similar in style and difficulty to HW problems. All proofs are short.
• Messy solutions will cause point deductions! It is not enough to have the correct idea, you have to be able to communicate it clearly, precisely, and concisely.
• To study, I recommend:
• Quickly review your notes and read the book. The most important parts is to review the definitions and theorems. If necessary, write those in a different sheet of paper so that you don't have to browse your book every time you need to refresh your memory.
• Also, review HW problems, as you might have seen, we often used them in other proofs. (Write them in a sheet of paper if you need to.)
• Work on HW10. The solutions are already posted, but try to do them without looking at the solution first!
• Look at all examples and solved problems.
• Review your old HWs. Redo all problems that you've missed. (You should always learn from your mistakes!) Always look for exactly what went wrong so that you do not repeat the same mistake!
• Look in solutions of solved problems for ideas that can make a proof shorter or simpler.
• Do as many problems as possible from the book.
• Work on problems from other sources, such as the additional bibliography given below.
• (09/09 - 7:40am) Change in make up work policy: As recommended by the university administration, I will not require a doctor's note in order to make up for missed work anymore. The change was suggested to avoid encouraging sick students to come to class while still contagious, and to prevent the Student Health Center to receive more appointments that necessary.

• (08/16 - 7:00pm) Please, check this section often. I will put announcements and important info here.

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### Instructor Contact and General Information

Instructor: Luís Finotti

Office: Aconda Court 211-H

Office Hours: MW 9:00am-10:00am or by appointment (subject to change!!)

Textbook: J. Rotman, ``A First Course In Abstract Algebra'', 3rd Edition, Prentice Hall, 2006.

Prerequisite: Math 300 (and 251/257).

Class: MWF 10:10-11:00 at HBB 103. (Section 1.)

Exams: Midterms: 09/30 (Wed) and 11/04 (Wed), in regular classroom and time; Final: 12/09 (Wed) from 10:15pm to 12:15pm, also in our regular classroom.

Grade: 25% for HW (quizzes and homework) + 20% for each Midterm + 35% for the Final. Note the weight of the quizzes/HWs!

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### Exams:

#### Midterm I

Here is the Midterm I and the Midterm I with solutions. Here is the Take Home Exam I and the Take Home Exam I with solutions.

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#### Midterm II

The exams were graded and returned in class on Thursday (11/06). (You can check your grades and statistics in Blackboard.) If you were not in class that day, you have to come by my office to pick up yours. (I will not take them to class anymore.)

Here is how I would curve these grades and the number of people with each grade:

 Grade Range # of people A 85-100 3 A- 83-84 1 B+ 79-82 3 B 74-78 3 B- 70-73 0 C+ 65-69 2 C 60-64 2 C- 55-59 0 D+ 50-54 0 D 45-49 0 D- 40-44 1 F < 40 3

Note: This curve is just to give you an idea of how I curve. In the end I will not even look at this curve, I will just curve all the final averages.

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#### Final

Here is the final: Final, Final Solution.

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### Course Information

#### Course Description

This course is a one-semester introduction to Abstract Algebra. (Math 455/456 and 457/458 are year long courses on the same subject, and hence cover much more.) The emphasis will be given to integers and polynomials, which are examples of commutative rings. The other main topic to be covered (at least superficially) is groups.

This course might be a bit of a shock to many students, as up to now most will not have dealt with discrete, rather than continuous (in the calculus sense) structures and proofs, which is what you usually deal with in calculus, differential equations, and when working with real numbers. So, it might take a little time for you to get use to the ideas and techniques used in this course.

Being an upper level course for math majors, most of the course will be spent on proofs (as in Math 300), and you will have to read and write many. I will assume you are comfortable doing both. We will also deal with induction and set theory (again from Math 300.) Other than that, there is really very little in terms of background knowledge necessary.

#### Course Structure

I will try to cover most of chapters 1 to 3, while skipping material with which you should be familiar (from Math 300), such as parts of sections 1.2 and 2.1. I might also introduce topics that are not necessarily covered in the text, or maybe from later sections. Hence, be aware that I will not necessarily follow the book too closely, but I will do my best to not deviate too much. I might, though, cover Chapter 3 (Ring Theory) before Chapter 2 (Group Theory), as I find it would be better to make sure we have enough time to cover rings well. (Our text was designed to allow that.)

#### Homework and Quizzes

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 due on Wednesday, when you will either turn in your HW or take a quiz on problems taken directly from the HW set due on that day. In the former case, two or three problems will be graded and count the same as a quiz. In the the latter, the HW will not be collected at all. You will not known in advance if the HW will be collected or if there will be a quiz, and hence you should be prepared for either one.

The quizzes will take place at the beginning of the class. You will not have extra time if you are late. You will have only ten or fifteen minutes to take a quiz, and so if you hadn't already solved the HW problems, you might not have enough time to come up with a solutions.

Points will be taken from messy solutions in all assignments, and you need to show work in all questions (unless stated otherwise)! (This same applies to quizzes, HWs, exams and all graded work!)

Calculators will not be allowed! (This includes HW, quizzes, and exams!)

I will do my best to post solutions to the most difficult problems. 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 greater than 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.

Finally, it is your responsibility to keep all your graded Quizzes, HW, and Midterms! It is very important to have them in case there is any problem with your grade. You can check all your scores at Blackboard. (Blackboard will be used only for scores. This is the official site for the course.)

#### Missed Work

There will be no make-up quizzes or exams. If you miss a quiz or exam and have a properly documented reason or due to sickness, your final will be used to make-up your score.

#### E-Mails

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, 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 a week or less, and it will be considered an official communication.

#### Feedback

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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### Legal Issues

#### Conduct

All students should be familiar and maintain their Academic Integrity: from
Hilltopics 2009/2010, pg. 40:

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 2009/2010, 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.

#### Disabilities

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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Here are some other books you might find helpful:

First of all, one should be careful that most books start with groups, not rings. The first two do, so might be good references. On the other hand, in most cases one can just skip to the sections on rings and find very few references to groups, so the others are also helpful.

Hungerford was my first option as the text for this course, but was too expensive. Herstein's ``Abstract Algebra'' was also considered as a text.

Fraleigh and Gallian are easy to ready and very complete, while Artin and Herstein's ``Topics in Algebra'' are of a higher level, and are often times used in honors courses.

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### Important dates:

Wednesday, August 19 - Classes begin.

Friday, August 28 - Last day to add, change grade options, or drop a full semester course without a "W".

Monday, September 07 - Labor Day Holiday. (No class.)

Wednesday, September 30 - Midterm I.

Thursday-Friday, October 15-16 - Fall Break. (No class.)

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

Wednesday, November 04 - Midterm II.

Tuesday, November 10 - Last day to drop with a WP/WF.

Thursday-Friday, November 26-27 - Thanksgiving. (No class.)

Tuesday, December 01 - Last Class Day.

Wednesday, December 09 - 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 to the harder problems, 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!)

CHANGE LOG:

• (11/29 - 10:35am) Added solutions for HW10.
• (11/18 - 4:55pm) Added solutions for HW9.
• (11/12 - 9:20am) Added solutions for HW8.
• (10/30 - 12:40pm) Added solutions for HW6 and HW7.
• (10/14 - 3:55pm) Added solutions for HW5.
• (09/26 - 9:10am) Added solutions for HW3 and HW4.
• (09/18 - 12:45pm) Added solutions for HW2.
• (09/03 - 10:45am) Posted solutions for HW1.

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### Homework

HW1 - Due on Wednesday 09/02 (note it was postponed!):

Section 1.3: 1.46, 1.50, 1.53, 1.57, 1.58, 1.60, 1.62. (Note that 1.53 and 1.55(i) were postponed to next week and 1.58 and 1.62 were added.)

HW2 - Due on Wednesday 09/16 (note it was postponed!):

Section 1.3: 1.53, 1.55 (i).

Section 1.4: 1.68, 1.69(i), 1.70(i), 1.71, 1.75.

HW3 - Due on Wednesday 09/23:

Section 1.5: 1.79, 1.81, 1.82(i), 1.85, 1.86, 1.87, 1.88.

HW4 -Do not turn in, but solve them to practice for the exam:

Section 1.5: 1.77, 1.78(ii), (iii), (iv), 1.91, 1.95.

HW5 - Due on Wednesday 10/14 (note it was postponed):

Section 3.1: 3.1 except (v) and (viii), 3.3(i) and (iii), 3.5, 3.6, 3.12, 3.13.

HW6 - Due on Wednesday 10/28 (note it was postponed):

Section 3.2: 3.17 except (v) and (vi), 3.19, 3.20, 3.26.

HW7 - Do not turn in, just practice for the exam:

Section 3.3: 3.29, 3.30, 3.32, 3.35 (if you are not familiar with complex numbers, replace them with real numbers, i.e., take alpha to be real), 3.37 (you can use 3.36 without proving it).

Section 3.5: 3.56(i), (ii), (iv), (vi), (vii).

HW8 - Due on Wednesday 11/11:

Section 3.5: 3.56(iii), (v), 3.58, 3.62, 3.64, 3.67.

HW9 - Due on Wednesday 11/18:

Section 3.7: 3.86 except (i), 3.87 except (ii), (v), (vii), (viii), 3.90(i), 3.91.

HW10 - Do not turn in, just practice for the final:

Section 2.2: 2.21, 2.22, 2.23, 2.25, 2.26, 2.27, 2.34. (Note that I took off 2.29, which is too long.)

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 lfinotti@utk.edu, and I'll update it and let you know.

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