## Syllabus for Number Theory (Math 5653), Spring 2014

### MW 2-3:15 pm, Clark 101.

• Instructor: Mikhail Ershov
• Office: Kerchof 302
• e-mail: ershov at virginia dot edu
• Office hours: 2 hrs 30 min TBA and by APPOINTMENT
• Course webpage:   http://people.virginia.edu/~mve2x/5653_Spring2014
Text: Elementary Number Theory, Gareth Jones and Josephine Jones, corrected edition.

### Prerequisites

• 1. You should have taken MATH 3354 and MATH 1320 prior to this class (note that MATH 3354 is an official prerequisite for MATH 5653, while MATH 1320 is a prerequisite for MATH 3354).
• 2. You should be comfortable with basic analytic techniques (mainly convergence and divergence tests for series and improper integrals) at the level of MATH 1320.
• 3. You should be familiar (and fairly comfortable) with basic group theory and you should have seen the very basic notions of ring and field theory (fields, rings, ideals).
• 4. You should have substantial prior exposure to proofs. You should be completely comfortable with the basic proof techniques (induction, contradiction, case exhaustion etc.) and have experience in both reading (and understanding) proofs and writing your own proofs.
If you are not comfortable with proofs, you will likely find this course very challenging. Prerequisites 3 and 4 should normally be satisfied if you took MATH 3354.

### Course outline

The course will start with a brief review of very basic concepts and results in number theory (greatest common divisor, prime factorization, congruences) which should be familiar to you from Math 3354. This material corresponds roughly to the first 3 chapters of the book and will be covered in the first 4 classes. We will spend approximately 9 more weeks on the material discussed in the textbook -- the plan is to cover the majority of the remaining topics, including techniques for solving congruences, the structure of the group of units of Zn, quadratic reciprocity, Riemann Zeta function, representation of integers by sums of squares and Fermat's theorem. In the remaining 2 weeks (these will not necessarily be the last 2 weeks of classes) we will cover additional topics including continued fractions, Pell's equation and a brief introduction to p-adic numbers.

### Evaluation

The course grade will be based on homework, two midterms and the final, with weights distributed as follows:
• homework: 30%
• midterms: 20% each
• final: 30%

### Exams

The final will be given in-class on Mon, May 6th, 2-5pm. The format of the midterms is open at this point, but most likely one of them will be given in class (during our regular class time), while the other will be take-home. It is possible that one of the midterms will have both take-home and in-class parts. The midterm dates given below are tentative and may be changed later.
• First midterm exam: Wed, Feb 26th
• Second midterm exam: Wed, Apr 9th
• Final exam: Sat, May 3rd, 2-5pm

### Make-up policy

• Make-ups or extensions (for take-home exams) will be given only under extreme circumstances (such as serious illness). Except for emergencies, you must obtain my permission for a make-up (resp. extension) before the exam (resp. due date).
• If you miss an in-class exam or fail to submit a take-home exam by the due date without a compelling reason, you will be assigned the score of 0 on that exam.
• University regulations specifically prohibit early make-ups.

### Homework

• Homework will be assigned weekly and will usually be due on Wednesdays.
• No late homework will be accepted. However, two lowest homework scores will be dropped.
• GRADING: it will not be possible to grade all homework problems. In each assignment 3-4 selected (but not announced in advance) problems will be graded for credit.

### Collaboration policy.

• On homework: you are welcome (and even encouraged) to work on homework together, but you must write up solutions independently, in your own words. In particular, you should not be consulting others during the process of writing down your solution.
• On take-home exams: you MAY NOT discuss problems with other people or use any resources (including web) except for the class textbook and your class notes. You may ask me questions about the exam problems, but normally I will only provide very brief hints.

### Announcements

Major announcements will be made in class and also posted on the course webpage. Some other announcements may only be made by e-mail, so check your e-mail account regularly.