A cubic ring of integers with the smallest Pythagoras number

TL;DR

This paper proves that the ring of integers in a specific totally real cubic field has the smallest Pythagoras number of 4 among such fields, and constructs a universal quadratic form based on this finding.

Contribution

It establishes the minimal Pythagoras number for totally real cubic fields and constructs a corresponding universal quadratic form.

Findings

Pythagoras number of the ring of integers is 4

Identifies which numbers are sums of squares in the field

Constructs a diagonal universal quadratic form in five variables

Abstract

We prove that the ring of integers in the totally real cubic subfield $K^{(49)}$ of the cyclotomic field $\mathbb{Q}(\zeta_7)$ has Pythagoras number equal to $4$. This is the smallest possible value for a totally real number field of odd degree. Moreover, we determine which numbers are sums of integral squares in this field, and use this knowledge to construct a diagonal universal quadratic form in five variables.