Pythagoras numbers for infinite algebraic fields

TL;DR

This paper investigates the Pythagoras numbers of rings of integers in various infinite algebraic fields, establishing conditions under which these numbers are finite or infinite, thus advancing understanding of sum-of-squares representations.

Contribution

It proves the infinitude of Pythagoras numbers for certain infinite fields and constructs examples with finite Pythagoras numbers, revealing new distinctions among algebraic fields.

Findings

Pythagoras number of the ring of integers in the compositum of all real quadratic fields is infinite.

Certain infinite totally real cyclotomic fields also have infinite Pythagoras numbers.

Constructed infinite degree totally real algebraic fields with finite Pythagoras numbers (1, 2, 3, at least 4).

Abstract

We prove that the Pythagoras number of the ring of integers of the compositum of all real quadratic fields is infinite. The same holds for certain infinite totally real cyclotomic fields. In contrast, we construct infinite degree totally real algebraic fields whose rings of integers have finite Pythagoras numbers, namely, one, two, three, and at least four.