In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard ord ...
.
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 ''Sentences'' (. ) is a compendium of Christian theology written by Peter Lombard around 1150. It was the most important religious textbook of the Middle Ages.
Background
The sentence genre emerged from works like Prosper of Aquitaine's ...
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 geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
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
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least on ...
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, every element of is a root of a non-zero polynomial with coefficients in . A field extens ...
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.
Def ...
.
[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 . In other words, a field is algebraically closed if the fundamental theorem of algebra ...
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