Factorization problems in class number two

Class-number Problems for Cubic Number Fields

The Gauss Class-Number Problems

In Articles 303 and 304 of his 1801 Disquisitiones Arithmeticae [Gau86], Gauss put forward several conjectures that continue to occupy us to this day. Gauss stated his conjectures in the language of binary quadratic forms (of even discriminant only, a complication that was later dispensed with). Since Dedekind’s time, these conjectures have been phrased in the language of quadratic fields. This...

On Gauss's Class Number Problems

Let h be the class number of binary quadratic forms (in Gauss's formulation). All negative determinants having some h = On ± 1 can be determined constructively: for h = 5 there are four such determinants; for h = 7, six; for A = 11, four; and for h = 13, six. The distinction between class numbers for determinants and for discriminants is discussed and some data are given. The question of one cl...

The factorization method for a class of inverse elliptic problems

In this paper the factorization method from inverse scattering theory and impedance tomography is extended to a class of general elliptic differential equations in divergence form. The inverse problem is to determine the interface ∂Ω of an interior change of the material parameters from the Neumann-Dirichlet map. Since absorption is allowed a suitable combination of the real and imaginary part ...

Continued Fractions and Class Number Two

We use the theory of continued fractions in conjunction with ideal theory (often called the infrastructure) in real quadratic fields to give new class number 2 criteria and link this to a canonical norm-induced quadratic polynomial. By doing so, this provides a real quadratic field analogue of the well-known result by Hendy (1974) for complex quadratic fields. We illustrate with several example...