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- Homework (Last updated: 02/16 at 1:45pm)
- Next Homework Due
- Blackboard: important announcements, grades, feedback, forums, calendar, etc.
- Piazza: our "Q&A - Math Related" for this course.
- Instructor Contact and General Info:
- Course Description and Information:
- Legal Issues:
- Additional Bibliography
- LaTeX
- Links
- 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 10:00 to 11:00 or by appointment. (Subject to change.) |
Textbook: | A. Frohlich and M.J. Taylor, "Algebraic Number Theory", 1st Edition, Cambridge University Press, 1993. |
Prerequisite: | Math 551/552, Math 651 recommended. |
Class Meeting Time: | TuTh 11:10-12:25 at Ayres 122. |
Exams: | None! |
Grade: | HWs only. |
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Course Information
Course Content
This course is a one-semester introduction to Algebraic Number Theory, a very important branch of mathematics.
Chapters and Topics
It is very tempting to cover the whole book! But that would be better suited for a year long sequence. My first instinct would be to covert Chapters I to VI, maybe skipping a few sections. It is a shame not to cover Chapter VIII, which uses powerful analytic tools that are an essential part of modern number theory. If the students are interested, we can make room somehow to cover some of its sections.
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!
You can do problems from our textbook or any other book in the listed Additional Bibliography below. If you want to do exercises from another text, please check with me first.
You will only need to do a few problems a week. To get an A, a minimum number of correct problems must be turned in by the due date.
If a problem you turn in is too far from correct, you might need to redo it in order to that problem to count towards the necessary number to get an A.
You also need to turn in your problems by the due date! I will allow one delayed HW per student only, unless there is a compelling reason to allow a second one.
It is your responsibility to keep all your graded HWs! It is very important to have them in case there is any problem with your grade.
I also urge all to read the material and keep up with the course! Even those less interested in the subject should be prepared enough to follow the lectures! This is essential to get the bare minimum out of the course!
Forums (Discussion Boards)
I've set up four forums on Blackboard: Math Related, Course Structure, Computer Related (questions about Sage, LaTeX, Blackboard, etc.) and Feedback (more on this last one 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!) You can access it through the link on the left panel of Blackboard or directly here: piazza.com/utk/spring2016/math652/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/math652.
When posting on Piazza, use the correct options. Choose a folder (there are folders for individual HWs, logistics 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.
Just like the Blackboard Forums, 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.
The 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!
Also, make sure to choose the appropriate forum for your question.
Please subscribe to all the forums to be notified of new posts! I will assume that everything posted on the forums was read by all!
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:
- D.A. Marcus "Number Fields",, 1977. Springer. -- An excellent book, that works out many examples concretely. It only suffers the unforgivable flaw of avoiding local methods entierely.
- J. Neukirch, "Algebraic Number Theory", 1999. Springer. -- Another excellent book, often used for this sort of courses.
- S. Lang, "Algebraic Number Theory", 2nd Ed., 1994. Springer. -- Perhaps a bit dry, but a good resource.
- Z.I. Borevich and I.R. Shafarevich. "Number Theory", 1966. Academic Press. Out of print! -- A classic and excellent book! You can find some scanned copies if you Google it.
- J.W.S. Cassels and A. Frohlich (editors), "Algebraic Number Theory", 1967. Academic Press. Out of print! -- Another classic! Quite hard to read (it is not quite a textbook), but you learn from the masters: Cassles, Hasse, Serre, Tate, Birch, Swinnerton-Dyer, Atiyah, etc. You can find some scanned copies if you Google it.
- W. Stein, "Algebraic Number Theory - A Computational Approach", 2012. Freely available! -- Also a good book, from which I plan to take some of the computational ideas.
- L.J. Goldstein "Analytic Number Theory", 1971. Prentice Hall. Out of print! -- Despite the title, it would be also a good choice for this course.
- J.S. Milne "Algebraic Number Theory", 2014. Freely Available! -- I am not very familiar with this one, but looks good and his books are indeed usually very good.
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LaTeX
This is not necessary to our class, but as graduates students, you really should learn some LaTeX. In any case, I leave it here in case someone wants to learn how type math, for instance to type their HW (or learn for when you write your thesis). 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):
- https://cloud.sagemath.com (This one is much more than just LaTeX)
- https://www.sharelatex.com
- https://www.overleaf.com
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 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
- Blackboard.
- Our Piazza Course Page ("Q&A - 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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Handouts
- Campus Syllabus.
- Exercises from Frohlich and Taylor's book.
- Errata 1: fix for a mistake in the proof of uniqueness of the representation of a finite commutative algebra into product of indecomposable algebras.
- Errata 2: a proof of the fact that an idempotent in a finite commutative algebra is the sum of primitive idempotents.
- Extra Notes 1: A proof of the fact that $K[\alpha] e_i(\alpha) \cong K[X]/f_i(X) \cdot K[X]$ [with the notation of the 02/12 lecture or as in page 26 of our text].
- Dual Basis for the Trace.
- Errata 3: fix for a mistake in the proof for the criterion for two $u$-norms to be equivalent.
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Homework
HW1 - Due on 02/23:
Choose 2 Problems between 1 to 11 from Chapter I (pg. 335), or similar problems from other texts. (You can find the exercises here.)
HW2 - Due on 03/24:
Choose 2 Problems between 1 to 5 or do only Problem 6 (counts as 2) from Chapter II (pg. 336), or similar problems from other texts. (You can find the exercises here.)
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