On the Pythagoras number of the simplest cubic fields
Type
Article
Departments
Annotation
Let ρ be a root of the polynomial x^3−ax^2−(a+3)x−1 where a≥3. We show that the Pythagoras number of the order Z[ρ] is equal to 6.
Trace and norm of indecomposable integers in cubic orders
Authors
Year
2023
Published
The Ramanujan Journal. 2023, 61(4), 1121-1144. ISSN 1382-4090.
Type
Article
Departments
Annotation
We study the structure of additively indecomposable integers in families of totally real cubic fields. We prove that for cubic orders in these fields, the minimal traces 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 or 2. We further give sharp upper bounds on the norms of indecomposable integers in our families.