Trace and norm of indecomposable integers in cubic orders

Abstract

We study the structure of the codifferent and of additively indecomposable integers in families of totally real cubic fields. We prove that for cubic orders in these fields, the minimal trace of indecomposable integers multiplied by totally positive elements of the codifferent can be arbitrarily large. This is very surprising, as in the so far studied examples of quadratic and simplest cubic fields, this minimum is 1 and 2. We further give sharp upper bounds on the norms of indecomposable integers in our families.

References in corpus (8)

• Number fields without n-ary universal quadratic forms
• Universal quadratic forms and elements of small norm in real quadratic fields
• Lifting problem for universal quadratic forms
• Universal quadratic forms, small norms and traces in families of number fields
• There are no universal ternary quadratic forms over biquadratic fields
• Norms of indecomposable integers in real quadratic fields
• Indecomposable integers in real quadratic fields
• Additive structure of totally positive quadratic integers

Cited by in corpus (5)

• Universal quadratic forms, small norms and traces in families of number fields
• Universal quadratic forms and indecomposables in number fields: A survey
• Sails for universal quadratic forms
• Arithmetic of cubic number fields: Jacobi-Perron, Pythagoras, and indecomposables
• Additive structure of non-monogenic simplest cubic fields