Pythagoras numbers of orders in biquadratic fields

Abstract

We examine the Pythagoras number $\mathcal{P}({\mathcal{O}}_K)$ of the ring of integers ${\mathcal{O}}_K$ in a totally real biquadratic number field $K$. We show that the known upper bound $7$ is attained in a large and natural infinite family of such fields. In contrast, for almost all fields $\mathbb{Q}(\sqrt{5},\sqrt{s})$ we prove $\mathcal{P}({\mathcal{O}}_K)=5$. Further we show that $5$ is a lower bound for all but seven fields $K$ and $6$ is a lower bound in an asymptotic sense.

44 pages. A minor correction: By mistake, we originally quoted another paper by M. Peters for the results on real quadratic fields

References in corpus (4)

• On Kitaoka's conjecture and lifting problem for universal quadratic forms
• On the Pythagoras number of the simplest cubic fields
• A cubic ring of integers with the smallest Pythagoras number
• On quadratic Waring's problem in totally real number fields

Cited by in corpus (7)

• On Kitaoka's conjecture and lifting problem for universal quadratic forms
• A cubic ring of integers with the smallest Pythagoras number
• Universal quadratic forms and indecomposables in number fields: A survey
• On quadratic Waring's problem in totally real number fields
• Additive structure of non-monogenic simplest cubic fields
• Arithmetic of cubic number fields: Jacobi-Perron, Pythagoras, and indecomposables
• Pythagoras numbers for infinite algebraic fields