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- Homework (Last updated: 04/28 at 10:30am)
- Next Homework Due
- Solutions to Selected HW Problems is now on Blackboard!
- Blackboard: important announcements, grades, feedback, forums, calendar, etc.
- Piazza (Math Related Forum).
- Instructor Contact and General Info:
- Course Description and Information:
- Legal Issues:
- Additional Bibliography
- LaTeX
- Links
- Videos
- Problems Likely To Be Assigned
- Handouts
Instructor Contact and General Information
Instructor: | Luís Finotti |
Office: | Ayres Hall 251 |
Phone: | 974-1321 (don't leave messages! -- me if I don't answer!) |
e-mail: | |
Office Hours: | TuTh from 10:00 to 11:00, or by appointment. (Subject to change). |
Textbook: | J. Rotman, "A First Course In Abstract Algebra", 3rd Edition, Prentice Hall, 2006. |
Prerequisite: | Math 300/307 (and 251/257). |
Class Meeting Time: | MWF 10:10-11 at Ayres 120. |
Exams: | Midterms: 02/24 (Wed) and 04/06 (Wed). |
Final: May 6 (Fri), from 8am to 10am. | |
Grade: | 25% for HW Average (lowest score dropped) + 20% for each midterm + 35% for the final. |
Note the weight of the HWs! |
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Course Information
Course Content
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 300/307), 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 300/307.) Other than that, there is really very little in terms of background knowledge necessary, except for matrices (Math 251/257), which will be used as examples.
Chapters and Topics
The goal would be to cover the following sections of our textbook (skipping some parts):
- Chapter 1:
- Section 1.3, 1.4: All.
- Section 1.4: All.
- Section 1.5: We will skip Example 1.78 (on pg. 70) until the end of the section. (It's quite interesting, but we don't have time.).
- Chapter 3:
- Section 3.1, 3.2: All.
- Section 3.3: The text gives a formal construction of polynomials here, but I will skip it and just treat them as the familiar objects that they are (or seem to be). Other than that, we will cover the whole section.
- Section 3.5: We will skip from Corollary 3.54 (pg. 259) to Theorem 3.63 (inclusive, on pg. 262) and from the subsection on Euclidean Rings (on pg. 267) until the end of the section.
- Section 3.6: We will skip from Lemma 3.87 (on pg. 278) to the end of the section.
- Section 3.7: If we are pressed on time, we might skip this section altogether (to get to Groups). If we do cover it, we will likely cover it all.
- Chapter 2:
- Section 2.2: All.
- Section 2.3: We will skip the subsection Symmetries (on pg. 137) to the end of the section.
- Section 2.4: If time allows, we should cover it all.
Sections 1.1, 1.2 and 2.1 are prerequisites. On Section 1.2 you all that come after Corollary 1.26 (on pg. 27), though.
Although not very likely, this outline is subject to change!
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.
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. (The list is likely incomplete, but chances of changing an assigned problem 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 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.
Points might be taken from messy solutions in all assignments, and you need to show work in all questions (unless stated otherwise)! This same applies to HWs, exams and all graded work.
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.
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.
I will post solutions to the most difficult problems on the section Solution to Selected HW Problems in Blackboard.
Finally, you can check all your scores at Blackboard.
Statements and Index Cards
I strongly recommend you write in a separate sheet of paper all definitions and statements of important theorems (lemmas, corollaries, propositions, etc.), and perhaps even a few more important examples that illustrate some technique. I recommend you do it before you start your HW on the corresponding section!
There are two main reasons for doing so: firstly, the act of writing helps you review and remember the main tools to solve problems in your HW. Secondly, having them on a separate sheet of paper makes it quicker to find what you need when doing your HW. (Hopefully by now you are aware that it is impossible to solve problems without knowing the relevant definitions and theorems!)
I would also recommend you write the definitions and theorems covered in class before the following class. This will help you follow better the new lecture. In fact, there is some benefit in writing them before they are introduced in class, as it makes easier to follow that lecture. But, in any event, you should do it before you do your HW.
Also, you can choose from this comprehensive list of definitions and theorems the most important (which doing the HW and following the lectures will help to rank) and put them in a 3" by 5" index card (front and back). I will collect those and will allow you to use your own in exams. (I will give them to you before the exams.) I will not check to see if there are any mistakes (thus, be extremely careful when writing, so that you don't have wrong statements when using them on exams!) and they are not worth any points, but they can help you during exams.
I was reluctant on allowing the use of index cards, since in my opinion you should study enough to know these definitions and theorems. But I also believe that it does help writing them: you have to look over all the statements and assess which are most important and write them again. Also, it allows you to spend more of your time, when preparing for exams, on the most important thing: solving problems!
Another warning: don't rely too much on these cards. Having the statements are not enough to know how to use them! It would take too much time for you to figure everything out from them. You need practice using them!
Note that these index cards should not be the same as the sheets of paper (with a more comprehensive list), as you will turn them to me, and I will only give them back to you on exams. (I will collect them back after the midterms, but you will keep them after the final.) If the index card is all that you do (and it shouldn't be!), at least make a copy (or take a picture) of it so you can use for future HWs or whenever you need a quick refresher.
You will turn these cards when HW is due, as described in the section Homework below. Only one index card will be allowed (unless mentioned otherwise) of the correct size. (You can cut a piece of paper in the correct size, 3" by 5", though.) The index card must have your name and the section it pertains clearly written on the top. It can be written on both sides, but it must contain only definitions and theorems from assigned section. I will not allow you to use your cards at all if you fail to follow these instructions! (The use of these cards is a privilege, not a right.) The cards must be turned in on the due date! Late cards will not be allowed.
When you turn in your first index card, you will turn it in inside a resealable small bag (e.g., a sandwich size zip lock bag) with your name on the bag. I will keep all your future index cards inside this same bag.
Finally, if there is interest, we could set up a collaboration to produce a complete collection of definitions and statements (and theorems). I could set up a LaTeX file (which is a program to typset math -- more on LaTeX below) in SageMathCloud (that allows us to share LaTeX documents, Google Docs style, and much more) that every student can write on, and together you can produce a complete, nicely formatted list (in PDF format!) for all to use. This does require the use of LaTeX, but I would be glad to help with the formatting and produce some templates in the beginning. If interested, we could divide the work among the students (or at least those who can use LaTeX, e.g., those who have taken Math 171).
Forums (Discussion Boards)
I've set up four forums on Blackboard:
- Math Related: In this forum you should ask questions about math only. (Actually set in Piazza.)
- Course Structure: In this forum you should ask questions about the course logistics, such as due dates, course policy, clarifications on topics from the Syllabus, etc.
- Computer Related: This is not very relevant to this course, but I've left it in case someone wants to explore Sage (a software that allows you to perform computations related to the course) or LaTeX. (More on that below.)
- Feedback: Feedback and comments about how the course is running. (More below.)
The Math Related Forum is actually set up on Piazza, since it allows the use of LaTeX, making it more convenient to discuss math. (Blackboard does take LaTeX, but the rendering is horrible and entering is slow!) Note that you don't have to use LaTeX at all to post there, or even know what LaTeX is! Even if you don't, I can use it, which makes my answers better to read, as it can properly display the math symbols. You can access it through the link on the left panel of Blackboard or directly here: piazza.com/utk/spring2016/math351/home. You should receive an invitation from me (by e-mail) to sign up for our Piazza class. If you don't, you can sign up here: piazza.com/utk/spring2016/math351.
When posting on Piazza, use the correct options. Choose a folder (there are folders for individual HWs, exams, lectures and other) and the proper settings from the post page (e.g., choose if it is a question or a note, etc.) To type in LaTeX, used double dollar signs ($$) instead of single ($) to surround your math equation.
Make sure you set your "Notifications Settings" on Piazza to receive notifications for all posts: Click on the gear on the top right of the Piazza site, the choose "Account/Email Setting", then "Edit Email Notifications" and then check "Automatically follow every question and note". Preferably, also set "Real Time" for both new and updates to questions and notes.
All other boards are hosted on Blackboard itself.
I urge you to use these forums often! If you are ever thinking of sending me an e-mail, think first if it could be posted in these forums. That way my answer might help others that have the same questions as you and will be always available to all. (Of course, if it is something personal (such as your grades), you should e-mail me instead.)
In all these forums you can post anonymously. (Just be careful to check the proper box!) But please don't post anonymously if you don't feel compelled to, as it would help me to know you, individually, much better.
Students can (and should!) reply to and comment on posts on the forums. Discussion is encouraged here! But please be careful with Math Related questions! You should not answer (or ask) questions about how to do a HW problem! (You can ask for hints or suggestions, though.) If you are uncertain if you can answer a (math related) post, please e-mail me first!
Also, make sure to choose the appropriate forum for your question.
Please subscribe to all the forums to be notified of new posts! (In Piazza and Blackboard!) I will assume that everything posted on the forums was read by all!
Missed Work
There will be no make-up exams. If you miss a HW or exam and have a properly documented reason, your final will be used to make-up your score.
More precisely: say you missed, say HW3, which involved sections 2.2 and 2.3, and say that in our final questions 3 and 4 are about the material of those sections. Then, the points you get in those questions of the final will make you HW3 grade.
Also, the lowest HW score will be dropped, to make up for any eventual problem for which you might not have a documented excuse.
E-Mail Policy
I will assume you check your e-mail at least once a day, but preferably you should check your e-mail often. 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.
Moreover, you should receive e-mails when announcements are posted on Blackboard, or when there is a new post in any of our forums. (Again, please subscribe to all of them, to receive notifications! Important information my appear in those.)
Feedback
Please, post all comments and suggestions regarding the course using Blackboard's Feedback. These can be posted anonymously (or not), just make sure to check the appropriate option. 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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Legal Issues
Conduct
All students should be familiar and maintain their Academic Integrity: from Hilltopics, pg. 19:
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. 74:
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!
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.
Campus Syllabus
Please, see also the Campus Syllabus.
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Additional Bibliography
Here are some other books you might find helpful:
- J. Fraleigh "A First Course in Abstract Algebra", 7th Ed., 2002. Addison Wesley.
- J. Gallian, "Contemporary Abstract Algebra", 7th Ed., 2009. Brooks Cole.
- M. Artin. "Algebra", 2nd Ed.,2011. Pearson.
- I. Herstein, "Topics in Algebra", 2nd Ed., 1975. Wiley.
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.
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LaTeX
This is not necessary to our class! I leave it here in case someone wants to learn how type math, for instance to type their HW. But again, you can ignore this section if you want to.
LaTeX is the most used software to produce mathematics texts. It is quite powerful and the final result is, when properly used, outstanding! Virtually all professional math text you will ever see is done with LaTeX, or one of its variants.
LaTeX is available for all platforms and freely available.
The problem is that it has a steep learning curve at first, but after the first difficulties are overcome, it is not bad at all.
One of the first difficulties one encounters is that it is not WYSIWYG ("what you see is what you get"). It resembles a programming language: you first type some code and then this code is processed to produce a nice document (a non-editable PDF file, for example). Thus, one has to learn how to "code" in LaTeX, but this brings many benefits.
I recommend that anyone with any serious interest in producing math texts to learn it! On the other hand, I don't expect all of you to do so. But note that there are processors that can make it "easier" to create LaTeX documents, by making it "point-and-click" and (somewhat) WYSIWYG.
Here are some that you can use online (no need to install anything and files are available online, but need to register):
- SageMathCloud (This one is much more than just LaTeX)
- ShareLaTeX
- Overleaf
The first one, SageMathCloud, is more than just for LaTeX, as you can also run Sage, which can do computations with the objects we will study in this course.
If you want to install LaTeX in your computer (so that you don't need an Internet connection), check here.
A few resources:
- TUG's Getting Started: some resources, from installation to first uses.
- A LaTeX Primer by D. R. Wilkins: a nice introduction. Here is a PDF version.
- Art of Problem Solving LaTeX resources. A very nice and simple introduction! (Navigate with the links under "LaTeX" bar on top.)
- LaTeX Symbol Lookup: Draw a symbol and the app will try to identify it and give you its LaTeX code.
- LaTeX Wikibook: A lot of information.
- LaTeX Cheat Sheet.
- Cheat Sheet for Math.
- List of LaTeX symbols.
- Comprehensive List of Math Symbols.
- Constructions: a very nice resource for more sophisticated math expressions.
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Links
- My web pages for Math 351 from Spring 2015 and Fall 2009. Includes all midterms and final with solutions.
- My web page for Math 506, which is a course similar to this one.
- Blackboard.
- Piazza (Math Related Forum).
- SageMathCloud
- UT Knoxville Home
- UTK's Math Department.
- Services for Current Students and MyUTK (registration, view your grades, etc.).
- Office of the Registrar
- Academic Calendars, including dates for add and drops, other deadlines, final exam dates, etc.
- Hilltopics.
- Office of Disability Services
- Office of Equity and Diversity (includes sexual harassment and discrimination).
- My homepage
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Videos
The videos below were made for a different course! So, if you watch them, you have to be careful with comments that I make about the course structure and what is important. They were made for Math 506 -- Algebra for Teachers. That course is taught online and its audience is teachers. Although we cover a lot of the same material, proofs are de-emphasized, unlike this course, where proofs are quite important!
On the other hand, I go over examples and solve problems, so it might be useful for you too. I also go over some computer programs, namely Sage and LaTeX, which are not part of our course, but you can learn them from the videos if you are interested.
If you are uncertain if something from the video is relevant or applies to our course, please ask! (Use a Blackboard Forum, please!)
Please let me know if you find any mistake in the videos!
- Introduction (Optional):
- Introduction to SageMathCloud: here I show a little about LaTeX and the use of SageMathCloud.
- Optional (but recommended): You can watch these videos on What is Algebra?, where I give a brief answer to the question, and on Algebraic Structures (made for Math 457), which goes over a few topics we will cover, but many others that we won't. I think these make a good introduction to our course.
- Section 1.3:
- Long division with negatives.
- Proof of the following Basic Lemma: Suppose that $d \mid a$. Then $d \mid (a+b)$ iff $d \mid b$.
- Euclid's Lemma.
- Example of the Extended Euclidean Algorithm.
- \(b\)-adic representation (1.53 with \(b=4\)).
- Problems: 1.46(ix), (x), 1.60.
- A few words about Example 1.49.
- Computations in Sage.
- Section 1.4:
- Section 1.5:
- General remarks on congruences.
- Powers in congruences.
- Corollary 1.65 (divisibility criteria for 3 and 9).
- Problems 1.77(vi), (vii), 1.78(ii) (and extra example), 1.91(i).
- CRT with non-relatively prime moduli.
- Computations with Sage.
- Section 3.1:
- General Remarks on rings.
- Examples of rings.
- Other operations (subtraction, multiplication by integers, powers).
- Integers modulo \(m\).
- Integral domains (and zero divisors).
- Subrings (including Gaussian integers, \(\mathbb{Q}[\sqrt{2}]\)).
- Units.
- Problem 3.2.
- Computations with Rings in Sage.
- Section 3.2:
- Fraction field.
- Problems 3.17(iv), (vii), 3.19, 3.27(i).
- Section 3.3:
- Watch this video about Section 3.3 (before reading the section!).
- Word about the derivative.
- Section 3.5:
- Section 3.7:
- Section 2.2:
- Section 2.3:
- Section 2.4:
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Handouts
- Campus Syllabus.
- Here is Section 1.3 from our textbook, in case you are still waiting for a copy from the bookstore.
- Here are my class notes on modular arithmetic (computations modulo $m$). (Watch out for mistakes!)
- Midterm 1 and solutions.
- Errata 1.
- Midterm 2 and solutions.
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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 1.3: 1.46, 1.47, 1.50, 1.53, 1.55(i), 1.57, 1.58, 1.60, 1.62.
Section 1.4: 1.68, 1.69(i), 1.70(i), 1.71, 1.75, 1.76(ii).
Section 1.5: 1.77, 1.78(ii), (iii), (iv), 1.79, 1.81, 1.82(i), 1.85, 1.86, 1.87, 1.88, 1.91, 1.95.
Section 3.1: 3.1 except (v) and (viii), 3.3(i) and (iii), 3.5, 3.6, 3.12, 3.13.
Section 3.2: 3.17 except (v) and (vi), 3.19, 3.20, 3.26.
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 -- also, this one is much easier with the tools from Section 3.5).
Section 3.5: 3.56(i)-(vii), 3.58, 3.62, 3.64, 3.67.
Section 3.7: 3.86 except (i), 3.87 except (ii), (v), (vii), (viii), 3.90(i), 3.91.
Section 2.2: 2.21 ((ii) is easier after Section 2.3), 2.22, 2.23, 2.25, 2.26, 2.27, 2.34 and this extra question.
Section 2.3: 2.36 (i) to (v) and (viii) to (ix), 2.37, 2.38, 2.40, 2.42, 2.44.
Section 2.4: 2.52, 2.54, 2.55, 2.56, 2.57.
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Solutions to Selected HW Problems
Moved to Blackboard!
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Homework
HW1 - Due on Wednesday 02/03:
Review: Read Sections 1.1 and 1.2 and review things you've forgotten from Math 300/307. From 1.2 you can skip complex numbers if you know the basics: sum, products, absolute value, inverses and geometric representation. We will also use matrices as examples in this course, so maybe a quick review of 251 might be a good idea, especially matrix operations (sums, products, etc.), determinants and inverses.
Section 1.3: 1.46 (i)-(v), (x), 1.47, 1.50, 1.57, 1.58, 1.60, 1.62.
Index Cards: (Don't forget your name in each and write the corresponding section; also bring it in a zip lock bag this first time!)
- One index card (3" by 5", front and back OK) for Sections 1.1 and 1.2.
- Two index cards for Section 1.3 up to page 48.
HW2 - Due on Wednesday 02/10:
Section 1.3: 1.46 (vi)-(ix), 1.53, 1.55(i).
Index Card: One index card (3" by 5") for the rest of Section 1.3 (from page 49 to the end).
HW3 - Due on Wednesday 02/17:
Section 1.4: 1.68, 1.69(i), 1.70(i), 1.71, 1.75, 1.76(ii).
Index Card: One index card for Section 1.4.
HW4 - Due on Wednesday 02/24 (because of the exam, you will only turn in the index card):
Section 1.5: 1.77(i)-(v), (viii), 1.79, 1.81, 1.82(i), 1.85, 1.86, 1.87, 1.88.
Index Card: One index card for Section 1.5 up to pg. 65. (Bring it for the exam!)
HW5 - Due on Wednesday 03/09:
Section 1.5: 1.77(vi), (vii), 1.78(ii), (iii), (iv), 1.91, 1.95.
Index Card: One index card for the rest of Section 1.5.
HW6 - Due on Wednesday 03/23:
Section 3.1: 3.3(i) and (iii), 3.5, 3.13.
Index Card: One index card for Section 3.1, up to pg. 224.
HW7 - Due on Wednesday 03/30:
Section 3.1: 3.1 except (v) and (viii), 3.6, 3.12.
Index Card: One index card for Section 3.1, from pg. 224 on.
HW8 - Due on Wednesday 04/06 (because of the exam, you will only turn in the index card):
Section 3.2: 3.17 except (vii) (note the change!), 3.19, 3.20, 3.26, 3.27 (added later!).
Section 3.3: 3.29 except (i), 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 -- also, this one is much easier with the tools from Section 3.5).
Index Card: One index card for Section 3.2 and one for 3.3.
HW9 - Due on Wednesday 04/20:
Section 3.5: 3.56(i)-(vii) (in (vii) it should say $k=\mathbb{F}_p$, not $k= \mathbb{F}_p(x)$), 3.58, 3.62, 3.64, 3.67 (here $R$ must be a domain! Otherwise the statement is false: for example $f=x (x-2)(x-4)$ is such that $x-4$ divides both $f$ and $f'$, but $f$ has no repeated roots. But note that we usually don't talk about GCDs if the ring of coefficients is not a domain or field).
Index Card: One card for both Sections 3.5 and 3.6. (Not one of each, one for both.)
HW10 - Not to be turned in! To practice for the final, work on:
Section 3.7: 3.86 except (i), 3.87 ((viii) is hard), 3.90(i), 3.91.
Section 2.2: 2.21 ((ii) is easier after Section 2.3), 2.22, 2.23, 2.25, 2.26, 2.27, 2.34 and this extra question.
Index Card: You can bring to the final one index card for Section 3.7 and one for Section 2.2 (and 2.1, if you want).
PLEASE, HIT "REFRESH" (OR "RELOAD") IN YOUR BROWSER WHEN VISITING THIS PAGE! 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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