Maximally elastic quadratic fields

Abstract

Recall that for a domain R where every nonzero nonunit factors into irreducibles, the elasticity of R is defined assup⁡{sr:π1⋯πr=ρ1⋯ρs, with all πi,ρj irreducible}. We call a quadratic field Kmaximally elastic if the ring of integers of K is a UFD and each element of {1,32,2,52,3,…}∪{∞} 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(2) is universally elastic, and we provide evidence for a conjectured characterization of maximally elastic quadratic fields.