Undergraduate Seminar in Number Theory 0366-3328-01
Tel Aviv, Fall 2021
Zeev Rudnick
The seminar will consist of presentations by the students of
various topics in Number Theory and its applications.
There will be one formal meeting per week with the entire class
as well as one-on-one meeting with the
instructor to prepare the lectures, make up a homework assignment
which will later be graded by the speakers and discuss the homework
grades.
Schedule
Wednesday 12-14, Schreiber 7
Syllabus
The main subject will be the arithmetic of the ring of integers of imaginary quadratic fields, applications of unique factorization, connection with the theory of binary quadratic forms,
and finally an extension to a noncommutative setting, of the quaternions.
In detail:
The Euclidean algorithm in imaginary quadratic fields:
Review: The Gaussian integers Z[i], the Euclidean algorithm and primes of the form x2+y2 (iff p=2 or p=1 mod 4)
The ring of integers in an imaginary quadratic number field. Units. Ideals. Irreducibles vs primes. PID implies irreducible=prime, and unique factorization. The norm of an ideal.
General Euclidean rings (see Herstein section 3.7).
The Eisenstein integers Z[(-1 +√-3)/2] and primes of the form x2+3y2 (iff p=3 or p=1 mod 3) (see Ireland & Rosen chapter 9 §1)
Other examples: Primes of the form =x2+2y2 iff p=2 or p=1,3 mod 8
Binary quadratic forms. See course notes and Chapter 6 of Buelle’s book.
Equivalence, strict equivalence, discriminant.
Reduction theory. The upper half-plane and the fundamental domain of the modular group.
Correspondence of ideals with binary quadratic forms.
New examples of imaginary quadratic fields of class number one, hence unique factorization: d=-19,-43,-67, -163.
Applications to representing p=x2+ny2
e.g. x2+163y2 iff (-163/p)=+1↔ (p/163)=+1 (for p≠ 2, 163).
Prime-Producing Polynomials
Rabinowitsch (1913): n2+n+A is prime for all n=0,1,…,A-2 if (and only if ) d=1-4A is squarefree and the ring of integers of Q( √ d) has unique factorization. Source: Fendel .
Integral quaternions and the four-square theorem (from Herstein chapter 7)
Definition and properties of the Hamilton quaternions Q=R[1,i,j,k] – an associative noncommutative division ring. Adjoints, norm.
The Hurwitz quaternions H=Z[i,j,k, (1+i+j+k)/2]. The units of H. The left-division algorithm in H. Consequence: every left ideal in H is principal.
Lagrange’s 4 square theorem: Every positive integer is a sum of 4 perfect squares.
Prerequisites:
All students need to have already completed the course
Introduction to Number Theory. Also useful would be the course Algebra B1.
Grading policy:
The presentation accounts for 75% of the final grade.
Attendance is mandatory and will be 10% of the final grade.
Weekly homework will be assigned and will contribute 15% of the final course
grade. Turning in the homework by the due date is mandatory.