Math 556: Number Theory II - Spring 2011

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## Important Notes

• (01/20 - 4:00pm) 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: Ayres Hall 243

Office Hours: MW 10am-11am or by appointment (subject to change!!)

Textbook: K. Ireland and M. Rosen, ``A Classical Introduction to Modern Number Theory'', 2nd Edition, 1990, Springer.

Prerequisite: One year of undergraduate abstract algebra and 555 (both recommended).

Class: TTh 11:10-12:25 at Ayres 112. (Section 1.)

Exams: This course will have no exams.

Grade: Only HWs. (More details to be discussed in class.)

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### Course Information

#### Course Description

This will be the continuation of 555. I will assume you remember what you've done, but will ``refresh your memory''. But, in fact, we might not need too much from 555. As in 555, you should be familiar with groups, rings, and fields at the undergraduate level. The course will 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, 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 more details on those if needed.

Here are the topics I'd like to cover: Zeta function (over finite fields), basics of Algebraic Number Theory, Dirichlet's Theorem of Primes in Arithmetic Progression, some Diophantine Equations, and basics of Elliptic Curves.

#### Course Structure

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 what I hope to cover:

• Chapter 10: Sections 1 and 2. Most of section 3 was done already.
• Chapter 11: Section 1-3 for sure, maybe 4 and 5.
• Chapter 12: All sections.
• Chapter 13: All sections. (Maybe skip 3.)
• Chapter 16: All sections.
• Chapter 17: Sections 5, 6, 8, 9.
• Chapters 18 and 19: As much as possible.

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.

#### Homeworks

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 collected on Thursdays. Each HW will have problems from the previous week (Tuesday and Thursday lectures). The problems to be turned in, as well as due dates, will be clearly posted here. I will suggest a few problems and ask you to turn in two or three problems a week.

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

In my opinion, doing the HW is one of the most important parts of the learning process, so I strongly recommend you work 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 can check all your scores at Blackboard. (Blackboard will be used only for scores. This is the official site for the course.)

#### 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, 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.

#### 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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Since this semester we will do bits of topics that could take a whole semester (or year!) to cover, there are many references, but not a single one (besides our text) which has exactly all we will cover. As usual, at the end of each section it gives references to other texts.

In any case, here are a few standard books:

• Algebraic Number Theory: by Fröllich and Taylor, Neukirch, Marcus, Borevich and Shafferevich, Lang.
• Elliptic Curves: Silverman (two volumes), Husemöller, Silverman and Tate, Cassels, Milne.
• Analytic Number Theory and Zeta Function: Apostol (undergraduate and graduate), Ivic, Patterson, Edwards, Stopple.

### Legal Issues

#### Conduct

All students should be familiar and maintain their Academic Integrity: from
Hilltopics 2010/2011, pg. 41:

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 the student present his or her own work in all academic projects, including tests, papers, homework, and class presentation. When incorporating the work of other scholars and writers into a project, the student must accurately cite the source of that work.

All students should follow the Honor Statement: from Hilltopics 2008/2009, 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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### Important dates:

Wednesday, January 12 - Classes begin.

Monday, January 17 - Martin Luther King, Jr. Day. (No class.)

Friday, January 21 - Last day to add, change grade options, or drop a full semester course without a "W".

Monday-Friday, March 14-18 - Spring Break. (No class.)

Tuesday, March 21 - Last day to drop a full term course with a "W".

Tuesday, April 5 - Last day to drop with a WP/WF.

Friday, April 22 - Spring Recess. (No class.)

Friday, April 29 - Last Class Day.

More dates.

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### Handouts

• Here are some statements from Field/Galois. But please be careful with typos or mistakes!! Please check them carefully. If you find mistakes, please let me know ASAP.
• Syllabus.

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### Homework

HW1 - Due on Thursday 01/27:

Chapter 8: Choose 2 or 3. (Suggestions: 8, 15, 26.)

HW2 - Due on Thursday 02/17:

Chapter 12: Choose 2 or 3 from 1, 2, 9, 10, 13, 14, 17, 18, 19. (Suggestions: 18, 19, if you haven't seen this result before.)

HW3 - Due on Thursday 03/10:

Chapter 12: Choose 2 or 3 from 4, 5, 6, 7, 8, 11, 12, 28, 31. (Suggestions: 4, 28, 31.)

HW4 - Due on Thursday 04/07:

Chapter 12: Choose 2 or 3 from 3, 15, 16, 25, 26, 29, 30, 32, 33, 34. (Suggestions: 3, 15, 26, 30.)

HW5 - Due on Wednesday 05/04:

Chapter 13: Choose 2 or 3. (My suggestions haven't been very popular, so I'll save us all the trouble and let you choose.)

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

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