Syllabus for Number Theory (Math 5653), Spring 2013
MW 2-3:15 pm, Monroe 111.
Text: Elementary Number Theory, Gareth Jones and Josephine Jones,
- Instructor: Mikhail Ershov
- Office: Kerchof 302
- e-mail: ershov at virginia dot edu
- Office hours: Mon 5-6, Tue 2-4 AND by appointment
- Course webpage: http://people.virginia.edu/~mve2x/5653_Spring2013
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.
- 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.
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
3 classes. We will spend approximately 7 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 5 weeks (these will not necessarily be the last 5 weeks of
classes) we will cover some more advanced topics, most likely taken from
the following list (we will probably only have time for 2 or 3 of those).
Continued fractions and Pell's equation
Dirichlet' theorem on primes in arithmetic progressions
Introduction to algebraic number fields
Introduction to elliptic curves
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: 25%
- midterms: 20% each
- final: 35%
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 20th
- Second midterm exam: Wed, Mar 27th
- Final exam: Mon, May 6th, 2-5pm
- 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 will be assigned weekly and will usually be due on Wednesdays.
- Please STAPLE your homework.
- No late homework will be accepted. However, two lowest
homework scores will be dropped.
- GRADING: due to the large class size, 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.
- 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.
Major announcements will be made in class and also posted on the course
Some other announcements may only be made by e-mail, so check your
- Monday, January 28 -- Last day to ADD a course, elect the audit
change the grading option (grade or CR/NC), or establish an independent
- Tuesday, January 29 -- Last day to DROP a course (deletion
from the transcript)
- Monday, March 18 -- Last day to withdraw from a course