Universal Quadratic Forms, Small Norms, and Traces in Families of Number Fields

Abstract

Abstract We obtain good estimates on the ranks of universal quadratic forms over Shanks’ family of the simplest cubic fields and several other families of totally real number fields. As the main tool, we characterize all the indecomposable integers in these fields and the elements of the codifferent of small trace. We also determine the asymptotics of the number of principal ideals of norm less than the square root of the discriminant.