In algebra, a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has a Pythagoras number equal to 1.
is an extension obtained by adjoining an element
containing it, unique up to isomorphism, called its Pythagorean closure.
there is an exact sequence involving the Witt rings where
denotes its torsion subgroup (which is just the nilradical of
[6] 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 (Iyanaga & Kawada 1980, 163 C).
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
[10] 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.
[11] This theorem states that if E/F is a finite field extension, and E is Pythagorean, then so is F.[12] As a consequence, no algebraic number field is Pythagorean, since all such fields are finite over Q, which is not Pythagorean.
[13] 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.
[12] The analogue of the Diller–Dress theorem holds: if E/F is a finite extension and E is superpythagorean then so is F.[14] 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.