Hereditarily non-pythagorean fields

TL;DR

The paper proves that for many fields, any proper finite extension of their pythagorean closure is not pythagorean, including number fields and certain finitely generated fields.

Contribution

It establishes a broad class of fields where finite extensions of the pythagorean closure lose the pythagorean property, extending previous understanding.

Findings

Finite extensions of pythagorean closures are not pythagorean in these fields.

Includes number fields and finitely generated fields over subfields.

Results apply to fields with transcendence degree at least one.

Abstract

We prove for a large class of fields $F$ that every proper finite extension of $F_{pyth}$, the pythagorean closure of $F$, is not a pythagorean field. This class of fields contains number fields and fields $F$ that are finitely generated of transcendence degree at least one over some subfield of $F$.