PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (191)
Orphanage
Unclass'd
Unproven (496)
Corrections (10)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
[parent] real closed fields (Example)

It is clear that the axioms for a structure to be an ordered field can written in $ L$, the first order language of ordered rings. It is also true that the following conditions can be written in a schema of first order sentences in this language. For each odd degree polynomial $ p \in K[x]$, $ p$ has a root.

Let $ A$ be all these sentences together with one that states that all positive elements have a square root. Then one can show that the consequences of $ A$ are a complete theory $ T$. It is clear that this theory is the theory of the real numbers. We call any $ L$ structure a real closed field.


The semi algebraic sets on a real closed field are Boolean combinations of solution sets of polynomial equalities and inequalities. Tarski showed that $ T$ has quantifier elimination, which is equivalent to the class of semi algebraic sets being closed under projection.


Let $ K$ be a real closed field. Consider the definable subsets of $ K$. By quantifier elimination, each is definable by a quantifier free formula, i.e. a boolean combination of atomic formulas. An atomic formula in one variable has one of the following forms:

  • $ f(x)>g(x)$ for some $ f,g \in K[x]$
  • $ f(x)=g(x)$ for some $ f,g \in K[x]$.
The first defines a finite union of intervals, the second defines a finite union of points. Every definable subset of $ K$ is a finite union of these kinds of sets, so is a finite union of intervals and points. Thus any real closed field is o-minimal.



"real closed fields" is owned by mathcam. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

See Also: theory

Also defines:  real closed field

This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: o-minimal, points, intervals, union, finite, variable, combination, quantifier free formula, subsets, definable, projection, closed under, class, quantifier elimination, inequalities, equalities, solution, combinations, Boolean, algebraic sets, real numbers, theory, complete theory, consequences, square root, positive elements, states, root, polynomial, degree, odd, language, sentences, first order, ordered rings, first order language, ordered field, structure, axioms, clear
There is 1 reference to this entry.

This is version 4 of real closed fields, born on 2003-01-23, modified 2007-07-10.
Object id is 3920, canonical name is RealClosedFields.
Accessed 3709 times total.

Classification:
AMS MSC: 03C64 (Mathematical logic and foundations :: Model theory :: Model theory of ordered structures; o-minimality)
 14P10 (Algebraic geometry :: Real algebraic and real analytic geometry :: Semialgebraic sets and related spaces)
 12D15 (Field theory and polynomials :: Real and complex fields :: Fields related with sums of squares )
 12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
Pending Errata and Addenda
None.
[ View all 3 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)