A cubic ring of integers with the smallest Pythagoras number

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.

8 pages; comments are welcome

References in corpus (2)

• Universal quadratic forms, small norms and traces in families of number fields
• Pythagoras numbers of orders in biquadratic fields

Cited by in corpus (8)

• Universal quadratic forms, small norms and traces in families of number fields
• Pythagoras numbers of orders in biquadratic fields
• 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
• Failures of integral Springer's Theorem