Schedule

Syllabus

In detail:

• Review: The Gaussian integers Z[i], the Euclidean algorithm and primes of the form x²+y² (iff p=2 or p=1 mod 4)
• Binary quadratic forms. Equivalence, strict equivalence, discriminant.
• Reduction theory. The upper half-plane and the fundamental domain of the modular group.
• Elementary genus theory, applications to representing p=x²+ny²
• The class number one problem, solution for even discriminants
• Prime-Producing Polynomials, Rabinowitsch's criterion (1913): n2+n+A is prime for all n=0,1,…,A-2 if and only if the class number h(1-4A)=1.

1. 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.
2. 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
• Correspondence of ideals with binary quadratic forms. (optional). New examples of imaginary quadratic fields of class number one, hence unique factorization: d=-19,-43,-67, -163. (optional)

Prerequisites:

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.

Homework

TBA

Office : Schreiber 308