Pythagoras numbers for infinite algebraic fields

TL;DR

The article investigates how the Pythagoras number $P(\\mathcal{O}_K)$ behaves for infinite totally real algebraic fields. It develops two complementary themes: infinite families with unbounded $P(\\mathcal{O}_K)$, shown for the compositum of all real quadratic fields and for real subfields of cyclotomic towers, and instances of infinite totally real fields with finite $P(\\mathcal{O}_K)$, obtained via explicit constructions and local-global considerations. The results hinge on trace methods and explicit basis/coefficient analyses to bound representations of totally positive elements as sums of squares, and together they motivate several open problems about the growth and finiteness of Pythagoras numbers in infinite settings. Overall, the paper advances understanding of how $P(\\mathcal{O}_K)$ can vary dramatically across infinite totally real extensions and sets a stage for further exploration of this arithmetic invariant.

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.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: see \ref{['prop:finite-PN']}
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Remark 2.3
• Lemma 3.1
• proof