Introduction to Number Theory
Prof. Zeev Rudnick
Fall 2022/23, Tel Aviv
Schedule: Lectures Monday 11-12, Wednesday 11-13, Checkpoint 001
Recitation sessions with Mr. Eden Kuperwasser, Monday 10-11 Schreiber 7 or Tuesday 14-15 Kaplun 118 and Mr. Roei Raveh Monday 10-11, Orenstein 110
Syllabus:
The course is an introductory course in basic number theory. It assumes
very little background beyond linear algebra and a solid course of
first year calculus.
The topics include
- The Euclidean algorithm, greatest common divisor, unique factorisation
into primes, linear Diophantine equations
- Continued fractions
- Congruences, the Chinese Remainder Theorem
- The multiplicative group of reduced residue classes modulo n,
Fermat's theorem
- The Euclidean algorithm for polynomials over a finite field
- Primitive roots
- Quadratic congruences, Legendre's symbol and quadratic reciprocity,
Jacobi's symbol
- Roots
of polynomial congruences, Hensel's lemma
- The Prime Number Theorem and its applications
- Primality testing
- Public Key Cryptography
- Pell's equation
- Diophantine approximation, Liouville's theorem on rational
approximations to algebraic numbers, Thue's equation
- Arithmetic in rings of integers of quadratic fields,
Euclidean rings,
sums of two squares and primes represented by binary quadratic forms.
Bibliography
Any introductory book on number theory will be useful. For example,
see:
-
"Elementary Number Theory" by W.J. LeVeque, Dover 1990.
- Elementary Number Theory, by D. Burton (available in Hebrew,
published by the Open University).
- A friendly introduction to number theory, by Joseph H. Silverman
(third edition, Prentice Hall)
-
Elementary Number Theory: Primes, Congruences and Secrets, by
William Stein (see online version).
- A more advanced text is "A classical introduction to modern number theory" by Ireland and Rosen
Attendance of lectures and recitation sessions is mandatory! We require presence in 80% of the lectures and 80% of the recitations
Homework:
This will be an important part of the course. In order to be eligible
to take the final exam, at least 50% of the assignments have to be
turned in on the week of their due date. 10% of the final grade will be
determined from the homework scores, which will be obtained as the average
grade of a certain number of assignments.
Contact: Zeev Rudnick, rudnick@tauex.tau.ac.il, Office : Schreiber 308
Teaching assistants: Eden Kupervwasser kuperwasser@tauex.tau.ac.il, Roei Raveh roeiraveh@mail.tau.ac.il
Course homepage: http://www.math.tau.ac.il/~rudnick/courses/int_numth.html