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 + λ2 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 Fpy containing it, unique up to isomorphism, called its Pythagorean closure. The Hilbert field is the minimal ordered Pythagorean field.
Every Euclidean field (an ordered field in which all positive elements are squares) is an ordered Pythagorean field, but the converse does not hold. A quadratically closed field is Pythagorean field but not conversely (R is Pythagorean); however, a non formally real Pythagorean field is quadratically closed.
The Witt ring of a Pythagorean field is of order 2 if the field is not formally real, and torsion-free otherwise. For a field F there is an exact sequence involving the Witt rings
where I W(F) is the fundamental ideal of the Witt ring of F and Tor I W(F) denotes its torsion subgroup (which is just the nilradical of W(F).
The following conditions on a field F are equivalent to F being Pythagorean:
Pythagorean fields can be used to construct models for some of Hilbert's axioms for geometry (Ito 1980, 163 C). The coordinate geometry given by Fn for F a Pythagorean field satisfies many of Hilbert's axioms, such as the incidence axioms, the congruence axioms and the axioms of parallels. However, in general this geometry need not satisfy all Hilbert's axioms unless the field F has extra properties: for example, if the field is also ordered then the geometry will satisfy Hilbert's ordering axioms, and if the field is also complete the geometry will satisfy Hilbert's completeness axiom.
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