Maximally elastic quadratic fields

Abstract

Recall that for a domain R where every nonzero nonunit factors into irreducibles, the elasticity of R is defined as$\sup \{\frac{s}{r}:{\pi }_1\cdots {\pi }_r={\rho }_1\cdots {\rho }_s,\text{ with all }{\pi }_i,{\rho }_j\text{ irreducible}\}.$ We call a quadratic field K maximally elastic if the ring of integers of K is a UFD and each element of $\{1,\frac{3}{2},2,\frac{5}{2},3,\dots \}\cup \{\infty \}$ appears as an elasticity of infinitely many orders inside K. This corresponds to the orders in K exhibiting, to the extent possible for a quadratic field, maximal variation in terms of the failure of unique factorization. Assuming the Generalized Riemann Hypothesis, we prove that $K=Q\left(\sqrt{2}\right)$ is universally elastic, and we provide evidence for a conjectured characterization of maximally elastic quadratic fields.