Universal quadratic forms, small norms and traces in families of number fields

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.

20 pages

References in corpus (3)

• On the Pythagoras number of the simplest cubic fields
• A cubic ring of integers with the smallest Pythagoras number
• Trace and norm of indecomposable integers in cubic orders

Cited by in corpus (7)

• On Kitaoka's conjecture and lifting problem for universal quadratic forms
• Number fields without universal quadratic forms of small rank exist in most degrees
• A cubic ring of integers with the smallest Pythagoras number
• On the Pythagoras number of the simplest cubic fields
• Universal quadratic forms and indecomposables in number fields: A survey
• Trace and norm of indecomposable integers in cubic orders
• Additive structure of non-monogenic simplest cubic fields