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 F is an extension obtained by adjoining an element √1 + λ for some λ in F. So a Pythagorean field is one closed under taking Pythagorean extensions. For any field F there is a minimal Pythagorean field F containing it, unique up to isomorphism, called its Pythagorean closure. The Hilbert field is the minimal ordered Pythagorean field.