Pythagorean field
Encyclopedia
In algebra, a Pythagorean field is a field
in which every sum of two squares is a square. A Pythagorean extension of a field F is an extension obtained by adjoining an element for some λ in F. So a Pythagorean field is one closed under
taking Pythagorean extensions. For any field there is a minimal Pythagorean field containing it, unique up to isomorphism, called its Pythagorean closure.
Pythagorean fields can be used to construct models for some of Hilbert's axioms
for geometry . The analytic 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.
The Pythagorean closure of a non-archimedean ordered field
, such as the Pythagorean closure of the field of rational function
s Q(t) in one variable over the rational numbers Q, 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 a non-Legendrian geometry and a semi-Euclidean geometry in which there are many lines though a point not intersecting a given line.
The Witt ring
of a Pythagorean field is of order 2 if the field is not formally real
, and torsion-free otherwise.
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
in which every sum of two squares is a square. A Pythagorean extension of a field F is an extension obtained by adjoining an element for some λ in F. So a Pythagorean field is one closed under
Closure (mathematics)
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...
taking Pythagorean extensions. For any field there is a minimal Pythagorean field containing it, unique up to isomorphism, called its Pythagorean closure.
Pythagorean fields can be used to construct models for some of Hilbert's axioms
Hilbert's axioms
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie , as the foundation for a modern treatment of Euclidean geometry...
for geometry . The analytic 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.
The Pythagorean closure of a non-archimedean ordered field
Non-Archimedean ordered field
In mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property. Examples are the Levi-Civita field, the hyperreal numbers, the surreal numbers, the Dehn field, and the field of rational functions with real coefficients with a suitable...
, such as the Pythagorean closure of the field of rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
s Q(t) in one variable over the rational numbers Q, 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 a non-Legendrian geometry and a semi-Euclidean geometry in which there are many lines though a point not intersecting a given line.
The Witt ring
Witt group
In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.-Definition:Fix a field k. All vector spaces will be assumed to be finite-dimensional...
of a Pythagorean field is of order 2 if the field is not formally real
Formally real field
In mathematics, in particular in field theory and real algebra, a formally real field is a field that admits an ordering which makes it an ordered field.-Alternative Definitions:...
, and torsion-free otherwise.