## Instructor Contact and General Information

 Instructor: Luís Finotti Office: Ayres Hall 251 Phone: 974-1321 (don't leave messages! -- e-mail (finotti at math dot utk dot edu) me if I don't answer!) e-mail: finotti at math dot utk dot edu Office Hours: MW 9-10 or by appointment.

 Textbook: K. Ireland and M. Rosen, "A Classical Introduction to Modern Number Theory", 2nd Edition, 1990, Springer. Prerequisite: one year of undergraduate algebra. Class Meeting Time: MWF 12:20am to 1:10pm. (Section 001.) Exams: No exams. Grade: Average of HWs, lowest score dropped.

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## Course Description and Information

### Course Content

This will be an introductory course in number theory. Very little background will be assumed, but you should be familiar with groups, rings, and fields at the undergraduate level. On the other hand, the course will be tailored to the students. Depending on the background of the audience, we will be able to cover more or even different topics. In particular, the knowledge of Galois Theory would be beneficial for the second semester, but it will not be assumed. Also, some knowledge of basic real and complex analysis will be assumed, again on an undergraduate level, but I will give much more details on those if needed.

We will start with very basic topics, such as factorization and congruences. I hope, depending on the audience, to move along those rather quickly, so that we can get to the more interesting parts soon, such as quadratic reciprocity, Gauss and Jacobi sums, equations over finite fields, and zeta function (for varieties over finite fields).

In fact, we have a few options. We could deal with more computational aspects, cryptography, analytic number theory... It depends on the interest of the class. The default will be to sample a little of everything.

### Chapters and Topics

As stated above, I will try to tailor the course to the students. This makes it harder to have a very precise plan, but here is the default:

• Chapter 1: all sections. (Quickly!)
• Chapter 2: all sections. (I will not rush through this chapter, as it contains new material.)
• Chapter 3: all sections. (Quickly, except maybe for section 4.)
• Chapter 4: all sections. (Not as quickly. Depends on the background of the class.)
• Chapter 5: all sections. (From here on, most of the material should be new, and I will slow down.)
• Chapters 6, 7 and 8: all sections. (I might go over sections 7.1 and 7.2, which introduces the basics of Finite Fields, quickly if the class have seen it before.)
• Chapter 10 and 11: all sections, as time allows.

This might be too ambitious. I might skip a few sections in between, or just give an overview in some cases. I find it difficult to really choose much here, as the topics are all very interesting to me, and I am eager to cover them all. But I will try to keep my enthusiasm from overloading you, and you should help me with that.

There are many good books in number theory. Not all cover all the topics covered by our text. The best source of alternative bibliography is Ireland and Rosen's book itself. At the end of each section it gives references to other texts. In any case, here are a few standard books:

### Homework Policy

Homework 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!

The HWs will "due" on Wednesdays. Each HW will have problems from the previous week (Monday, Wednesday and Friday lectures). I will not be very strict on dates, but I do not want you to turn in all your homework on the last day! I will also suggest a few problems, but you have the freedom to choose the ones you intend to do. As long as you turn in two to three (three if you choose really easy ones, please) a week, you should be fine. (Assuming you are getting most of the problems right.)

Of course, you get out what you put in, so students interested in algebra and number theory should at least try to do the harder problems, even if you do not turn them in. You are always welcome to come discuss problems with me.

In any event, no matter what you turn in, you should take a look at all exercises, as you will find interesting results and applications there. Sometimes we might need some results that were exercises from previous sections.

It's unlikely I will post solutions, but if you want to see some solution, you can come see me.

Besides HW, I would urge you to review the material often! I really don't want students to get lost in class, by forgetting theorems and definitions. Most of all, I want you to be able to follow the lectures! Of course, one way to make yourself keep up is to work on problems often, so try to not let your HWs be late often.

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 keep all your HW scores at Blackboard.

### Piazza (Discussion Board)

We will use Piazza for online discussions. The advantage of Piazza (over other discussion boards) is that it allows us (or simply me) to use math symbols efficiently and with good looking results (unlike Blackboard).

To enter math, you can use LaTeX code. (See the section on LaTeX below.) The only difference is that you must surround the math code with double dollar signs () instead of single ones (\$). Even if you don't take advantage of this, I can use making it easier for you to read the answers.

You can access Piazza through the link on the left panel of Blackboard or directly here: https://piazza.com/utk/fall2016/math555/home. (There is also a link at the "Navigation" section on the top of this page and on the Links section.)

To keep things organized, I've set up a few different folders/labels for our discussions:

• Homework: Each homework has its own folder. Ask question related to each chapter or exam in the corresponding folder.
• Course Structure: Ask questions about the class, such as "how is the graded computed", "when is the final", etc. in this folder. (Please read the Syllabus first, though!)
• Feedback: Give (possibly anonymous) feedback about the course using this folder.
• Other: In the unlikely event that your question/discussion doesn't fit in any of the above, please use this folder.

I urge you to use Piazza often for discussions! (This is specially true for Feedback!) If you are ever thinking of sending me an e-mail, think first if it could be posted there. 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.)

Note that 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 Piazza. Discussion is encouraged here!

Also, please don't forget to choose the appropriate folder(s) (you can choose more than one, like a label) for your question. And make sure to choose between Question, Note or Poll.

Also, you can send Private Messages in Piazza. So, if you have a math question not appropriate for the whole class (e.g., your question gives away the solution of a HW problem), you can send me a private message instead of an e-mail. That way my reply can have the math symbols nicely formatted.

Important: 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. I will consider a post in Piazza official communication in this course, I will assume all have read every single post there!

### 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 where there is a new post in Piazza. (Again, please subscribe to receive notifications in Piazza! Important information my appear in those.)

### Feedback

Please, post all comments and suggestions regarding the course using Piazza. Usually these should be posted as Notes and put in the Feedback folder/label (and add other labels if relevant). 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. 46:

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. 16:

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, Sexual Assault and Discrimination information, please visit the Office of Equity and Diversity.

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## LaTeX

This (LaTeX) is not necessary to our course! The only benefit for us is to help you post messages in Piazza with math symbols and type your homework. But, if you intend to write a thesis (or some math) at some point, might as well learn it now. But feel free to skip the rest of this section.

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):

If you want to install LaTeX in your computer (so that you don't need an Internet connection), check here.

A few resources:

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## Homework

HW1 - "Due" on Wednesday 08/31:

Chapter 1: Choose 2 or 3. (If you want suggestions: 12, 30, 38.)

HW2 - "Due" on Wednesday 09/28:

Chapter 2: Choose 2 or 3. (If you want suggestions: choose 2 or 3 from 13, 21, 26, 27.)

HW3 - "Due" on Thursday 10/05:

Chapter 3: Choose 2 or 3. (If you want suggestions: choose 2 or 3 from 11, 15 (note that the last term is 1/(p-1)), 17, 23, 25, 26.)

HW4 - Due on Thursday 10/19:

Chapter 4: Choose 2 or 3. (If you want suggestions: 5, 11, 22.)

HW5 - Due on Wednesday 11/02:

Chapter 5: Choose 2 or 3. (There are too many good problems in this section. If you really want suggestions: 26, 27, 28.)

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 week, write me an e-mail at lfinotti@utk.edu, and I'll update it and let you know.

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