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 formula_1 is an extension obtained by adjoining an element formula_2 for some formula_3 in formula_1. So a Pythagorean field is one closed under taking Pythagorean extensions. For any field formula_1 there is a minimal Pythagorean field formula_6 containing it, unique up to isomorphism, called its Pythagorean closure. The "Hilbert field" is the minimal ordered Pythagorean field.
Properties.
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 (formula_7 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 formula_1 there is an exact sequence involving the Witt rings
where formula_10 is the fundamental ideal of the Witt ring of formula_1 and formula_12 denotes its torsion subgroup (which is just the nilradical of formula_13).
Equivalent conditions.
The following conditions on a field "F" are equivalent to "F" being Pythagorean:
Models of geometry.
Pythagorean fields can be used to construct models for some of Hilbert's axioms for geometry . The coordinate geometry given by formula_14 for formula_1 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.
The Pythagorean closure of a non-archimedean ordered field, such as the Pythagorean closure of the field of rational functions formula_16 in one variable over the rational numbers formula_17 can be used to construct non-archimedean geometries that satisfy many of Hilbert's axioms but not his axiom of completeness. Dehn used such a field to construct two Dehn planes, examples of non-Legendrian geometry and semi-Euclidean geometry respectively, in which there are many lines though a point not intersecting a given line but where the sum of the angles of a triangle is at least π.
Diller–Dress theorem.
This theorem states that if "E"/"F" is a finite field extension, and "E" is Pythagorean, then so is "F". As a consequence, no algebraic number field is Pythagorean, since all such fields are finite over Q, which is not Pythagorean.
Superpythagorean fields.
A superpythagorean field "F" is a formally real field with the property that if "S" is a subgroup of index 2 in "F"∗ and does not contain −1, then "S" defines an ordering on "F". An equivalent definition is that "F" is a formally real field in which the set of squares forms a fan. A superpythagorean field is necessarily Pythagorean.
The analogue of the Diller–Dress theorem holds: if "E"/"F" is a finite extension and "E" is superpythagorean then so is "F". In the opposite direction, if "F" is superpythagorean and "E" is a formally real field containing "F" and contained in the quadratic closure of "F" then "E" is superpythagorean.