In algebra, a Pythagorean field is a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
in which every sum of two squares is a square: equivalently it has
Pythagoras number equal to 1. A Pythagorean extension of a field
is an extension obtained by adjoining an element
for some
in
. So a Pythagorean field is one
closed under
In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but n ...
taking Pythagorean extensions. For any field
there is a minimal Pythagorean field
containing it, unique
up to isomorphism Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
, called its Pythagorean closure.
[Milnor & Husemoller (1973) p. 71] The ''Hilbert field'' is the minimal ordered Pythagorean field.
[Greenberg (2010)]
Properties
Every
Euclidean field
In mathematics, a Euclidean field is an ordered field for which every non-negative element is a square: that is, in implies that for some in .
The constructible numbers form a Euclidean field. It is the smallest Euclidean field, as every Eu ...
(an
ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered field ...
in which all non-negative elements are squares) is an ordered Pythagorean field, but the converse does not hold.
[Martin (1998) p. 89] A
quadratically closed field is Pythagorean field but not conversely (
is Pythagorean); however, a non
formally real Pythagorean field is quadratically closed.
[Rajwade (1993) p.230]
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 there is an ]exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
involving the Witt rings
:
where is the fundamental ideal of the Witt ring of [Milnor & Husemoller (1973) p. 66] and denotes its torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...
(which is just the nilradical of ).[Milnor & Husemoller (1973) p. 72]
Equivalent conditions
The following conditions on a field ''F'' are equivalent to ''F'' being Pythagorean:
* The general ''u''-invariant ''u''(''F'') is 0 or 1.[Lam (2005) p.410]
* If ''ab'' is not a square in ''F'' then there is an order on ''F'' for which ''a'', ''b'' have different signs.[Lam (2005) p.293]
* ''F'' is the intersection of its Euclidean closure
In mathematics, a Euclidean field is an ordered field for which every non-negative element is a square: that is, in implies that for some in .
The constructible numbers form a Euclidean field. It is the smallest Euclidean field, as every Eu ...
s.[Efrat (2005) p.178]
Models of geometry
Pythagorean fields can be used to construct models for some of Hilbert's axioms
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book '' Grundlagen der Geometrie'' (tr. ''The Foundations of Geometry'') as the foundation for a modern treatment of Euclidean geometry. Other well-known modern ...
for geometry . The coordinate geometry given by for 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 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 ...
, such as the Pythagorean closure of the field of rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s in one variable over the rational numbers 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 In geometry, Max Dehn introduced two examples of planes, a semi-Euclidean geometry and a non-Legendrian geometry, that have infinitely many lines parallel to a given one that pass through a given point, but where the sum of the angles of a triangle ...
and semi-Euclidean geometry In geometry, Max Dehn introduced two examples of planes, a semi-Euclidean geometry and a non-Legendrian geometry, that have infinitely many lines parallel to a given one that pass through a given point, but where the sum of the angles of a triangle ...
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.[Dehn (1900)]
Diller–Dress theorem
This theorem states that if ''E''/''F'' is a finite field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
, and ''E'' is Pythagorean, then so is ''F''.[Lam (1983) p.45] As a consequence, no algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
is Pythagorean, since all such fields are finite over Q, which is not Pythagorean.[Lam (2005) p.269]
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''.][Lam (1983) p.47] 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.[Lam (1983) p.48]
Notes
References
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* {{citation , title=Squares , volume=171 , series=London Mathematical Society Lecture Note Series , first=A. R. , last=Rajwade , publisher=Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press
A university press is an academic publishing hou ...
, year=1993 , isbn=0-521-42668-5 , zbl=0785.11022
Field (mathematics)