Pythagorean field

In algebra, a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has Pythagoras number equal to 1. A Pythagorean extension of a field ${\displaystyle F}$ is an extension obtained by adjoining an element ${\displaystyle {\sqrt {1+\lambda ^{2}}}}$ for some ${\displaystyle \lambda }$ in ${\displaystyle F}$. So a Pythagorean field is one closed under taking Pythagorean extensions. For any field ${\displaystyle F}$ there is a minimal Pythagorean field ${\textstyle F^{\mathrm {py} }}$ containing it, unique up to isomorphism, called its Pythagorean closure.^[1] The Hilbert field is the minimal ordered Pythagorean field.^[2]