In
mathematics, in particular in
field theory and
real algebra, a formally real 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 ...
that can be equipped with a (not necessarily unique) ordering that makes it 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 ...
.
Alternative definitions
The definition given above is not a
first-order definition, as it requires quantifiers over
sets. However, the following criteria can be coded as (infinitely many) first-order
sentences
''The Four Books of Sentences'' (''Libri Quattuor Sententiarum'') is a book of theology written by Peter Lombard in the 12th century. It is a systematic compilation of theology, written around 1150; it derives its name from the ''sententiae'' o ...
in the language of fields and are equivalent to the above definition.
A formally real field ''F'' is a field that also satisfies one of the following equivalent properties:
[Milnor and Husemoller (1973) p.60]
* −1 is not a sum of
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length ad ...
s in ''F''. In other words, the
Stufe of ''F'' is infinite. (In particular, such a field must have
characteristic 0, since in a field of characteristic ''p'' the element −1 is a sum of 1s.) This can be expressed in first-order logic by
,
, etc., with one sentence for each number of variables.
* There exists an element of ''F'' that is not a sum of squares in ''F'', and the characteristic of ''F'' is not 2.
* If any sum of squares of elements of ''F'' equals zero, then each of those elements must be zero.
It is easy to see that these three properties are equivalent. It is also easy to see that a field that admits an ordering must satisfy these three properties.
A proof that if ''F'' satisfies these three properties, then ''F'' admits an ordering uses the notion of
prepositive cones and positive cones. Suppose −1 is not a sum of squares; then a
Zorn's Lemma argument shows that the prepositive cone of sums of squares can be extended to a positive cone . One uses this positive cone to define an ordering: if and only if belongs to ''P''.
Real closed fields
A formally real field with no formally real proper
algebraic extension
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ex ...
is a
real closed field
In mathematics, a real closed field is a field ''F'' that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.
D ...
.
[Rajwade (1993) p.216] If ''K'' is formally real and Ω is an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
containing ''K'', then there is a real closed
subfield of Ω containing ''K''. A real closed field can be ordered in a unique way,
[ and the non-negative elements are exactly the squares.
]
Notes
References
*
*
{{DEFAULTSORT:Formally Real Field
Field (mathematics)
Ordered groups
pl:Ciało (formalnie) rzeczywiste