Real closed field
In
Definitions
A real closed field is a field F in which any of the following equivalent conditions are true:
- F is
elementarily equivalent to the real numbers. In other words it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in F if and only if it is true in the reals. - There is a
total order on F making it an ordered field such that, in this ordering, every positive element of F is a square in F and any polynomial of odd degree with coefficients in F has at least one root in F. - F is a
formally real field such that every polynomial of odd degree with coefficients in F has at least one root in F, and for every element a of F there is b in F such that a = b or a = −b. - F is not
algebraically closed but its algebraic closure is a finite extension . - F is not algebraically closed but the
field extension is algebraically closed. - There is an ordering on F which does not extend to an ordering on any proper
algebraic extension of F. - F is a formally real field such that no proper algebraic extension of F is formally real. (In other words, the field is maximal in an algebraic closure with respect to the property of being formally real.)
- There is an ordering on F making it an ordered field such that, in this ordering, the
intermediate value theorem holds for all polynomials over F. - F is a field and a
real closed ring .
If F is an ordered field (not just orderable, but a definite ordering is fixed as part of the structure), the Artin–Schreier theorem states that F has an algebraic extension, called the real closure K of F, such that K is a real closed field whose ordering is an extension of the given ordering on F, and is unique up to a unique isomorphism of fields (note that every
If F is a field (so this time, no order is fixed, and it is even not necessary to assume that F is orderable) then F still has a real closure, which in general is not a field anymore, but a
Model Theory: decidability and quantifier elimination
The theory of real closed fields was invented by algebraists, but taken up with enthusiasm by logicians. By adding to the
one obtains a
Decidability means that there exists at least one
The decision procedures are not necessarily practical. The
The algorithm Tarski proposed for
Basu and Roy (1996) proved that there exists a well-behaved algorithm to decide the truth of a formula ∃x1,…,∃xk P1(x1,…,xk)⋈0∧…∧Ps(x1,…,xk)⋈0 where ⋈ is <, > or =, with complexity in arithmetic operations sd.
Order properties
A crucially important property of the real numbers is that it is an
The archimedean property is related to the concept of
We have therefore the following invariants defining the nature of a real closed field F:
To this we may add
These three cardinal numbers tell us much about the order properties of any real closed field, though it may be difficult to discover what they are, especially if we are not willing to invoke
The generalized continuum hypothesis
The characteristics of real closed fields become much simpler if we are willing to assume the
Moreover, we do not need ultrapowers to construct Ϝ, we can do so much more constructively as the subfield of series with a countable number of nonzero terms of the field
Ϝ however is not a complete field; if we take its completion, we end up with a field Κ of larger cardinality. Ϝ has the cardinality of the continuum which by hypothesis is
Examples of real closed fields
References
- Basu, Saugata, , and Marie-Françoise Roy (2003) "Algorithms in real algebraic geometry" in Algorithms and computation in mathematics. Springer. ISBN 3540330984 (online version)
- Caviness, B F, and Jeremy R. Johnson, eds. (1998) Quantifier elimination and cylindrical algebraic decomposition. Springer. ISBN 3211827943
- Chen Chung Chang and Howard Jerome Keisler (1989) Model Theory. North-Holland.
- Dales, H. G., and W. Hugh Woodin (1996) Super-Real Fields. Oxford Univ. Press.
- Mishra, Bhubaneswar (1997) "Computational Real Algebraic Geometry," in Handbook of Discrete and Computational Geometry. CRC Press. 2004 edition, p. 743. ISBN 1-58488-301-4
- Alfred Tarski (1951) A Decision Method for Elementary Algebra and Geometry. Univ. of California Press.
- J. Davenport and J. Heintz, "Real quantifier elimination is doubly exponential", Journal of Symbolic Computation 5:1–2 (1988), pp. 29–35.
External links
Retrieved from : http://en.wikipedia.org/wiki/Real_closed_field