Bose gas

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.

Definitions

A real closed field is a field F in which any of the following equivalent conditions are true:

1. 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 fieldsTemplate:Clarify is true in F if and only if it is true in the reals.
2. There is a total order on F making it an ordered field such that, in this ordering, every positive element of F has a square root in F and any polynomial of odd degree with coefficients in F has at least one root in F.
3. 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 = b² or a = −b².
4. F is not algebraically closed but its algebraic closure is a finite extension.
5. F is not algebraically closed but the field extension $F({\sqrt {-1}})$ is algebraically closed.
6. There is an ordering on F which does not extend to an ordering on any proper algebraic extension of F.
7. 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.)
8. 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.
9. F is a real closed ring.

If F is an ordered field (not just orderable, but a definite ordering P 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 P on F, and is unique up to a unique isomorphism of fields (note that every ring homomorphism between real closed fields automatically is order preserving, because x ≤ y if and only if ∃z y = x+z²). For example, the real closure of the rational numbers is the field $\mathbb {R} _{alg}$ of real algebraic numbers. The theorem is named for Emil Artin and Otto Schreier, who proved it in 1926.

If (F,P) is an ordered field, and E is a Galois extension of F, then by Zorn's Lemma there is a maximal ordered field extension (M,Q) with M a subfield of E containing F and the order on M extending P: M is the relative real closure of (F,P) in E. We call (F,P) real closed relative to E if M is just F. When E is the algebraic closure of F we recover the definitions above.

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 real closed ring. For example the real closure of the field $\mathbb {Q} ({\sqrt {2}})$ is the ring $\mathbb {R} _{alg}\times \mathbb {R} _{alg}$ (the two copies correspond to the two orderings of $\mathbb {Q} ({\sqrt {2}})$). Whereas the real closure of the ordered subfield $\mathbb {Q} ({\sqrt {2}})$ of $\mathbb {R}$ is again the field $\mathbb {R} _{alg}$.

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 ordered field axioms

• an axiom asserting that every positive number has a square root, and
• an axiom scheme asserting that all polynomials of odd degree have at least one root,

one obtains a first-order theory. Tarski (1951) proved that the theory of real closed fields in the first order language of partially ordered rings (consisting of the binary predicate symbols "=" and "≤", the operations of addition, subtraction and multiplication and the constant symbols 0,1) admits elimination of quantifiers. The most important model theoretic consequences hereof: The theory of real closed fields is complete, o-minimal and decidable.

Decidability means that there exists at least one decision procedure, i.e., a well-defined algorithm for determining whether a sentence in the first order language of real closed fields is true. Euclidean geometry (without the ability to measure angles) is also a model of the real field axioms, and thus is also decidable.

The decision procedures are not necessarily practical. The algorithmic complexities of all known decision procedures for real closed fields are very high, so that practical execution times can be prohibitively high except for very simple problems.

The algorithm Tarski proposed for quantifier elimination has NONELEMENTARY complexity, meaning that no tower $2^{2^{\cdot ^{\cdot ^{\cdot ^{n}}}}}$ can bound the execution time of the algorithm if n is the size of the problem. Davenport and Heintz (1988) proved that quantifier elimination is in fact (at least) doubly exponential: there exists a family Φ_n of formulas with n quantifiers, of length O(n) and constant degree such that any quantifier-free formula equivalent to Φ_n must involve polynomials of degree $2^{2^{\Omega (n)}}$ and length $2^{2^{\Omega (n)}}$, using the Ω asymptotic notation. Ben-Or, Kozen, and Reif (1986) proved that the theory of real closed fields is decidable in exponential space, and therefore in doubly exponential time.

Basu and Roy (1996) proved that there exists a well-behaved algorithm to decide the truth of a formula ∃x₁,…,∃x_k P₁(x₁,…,x_k)⋈0∧…∧P_s(x₁,…,x_k)⋈0 where ⋈ is <, > or =, with complexity in arithmetic operations s^k+1d^O(k). In fact, the existential theory of the reals can be decided in PSPACE.

Order properties

A crucially important property of the real numbers is that it is an Archimedean field, meaning it has the Archimedean property that for any real number, there is an integer larger than it in absolute value. An equivalent statement is that for any real number, there are integers both larger and smaller. Such real closed fields that are not Archimedean, are non-Archimedean ordered fields. For example, any field of hyperreal numbers is real closed and non-Archimedean.

The Archimedean property is related to the concept of cofinality. A set X contained in an ordered set F is cofinal in F if for every y in F there is an x in X such that y < x. In other words, X is an unbounded sequence in F. The cofinality of F is the size of the smallest cofinal set, which is to say, the size of the smallest cardinality giving an unbounded sequence. For example natural numbers are cofinal in the reals, and the cofinality of the reals is therefore $\aleph _{0}$.

We have therefore the following invariants defining the nature of a real closed field F:

• The cardinality of F.

• The cofinality of F.

To this we may add

• The weight of F, which is the minimum size of a dense subset of F.

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 generalized continuum hypothesis. There are also particular properties which may or may not hold:

• A field F is complete if there is no ordered field K properly containing F such that F is dense in K. If the cofinality of K is κ, this is equivalent to saying Cauchy sequences indexed by κ are convergent in F.

• An ordered field F has the η_α property for the ordinal number α if for any two subsets L and U of F of cardinality less than $\aleph _{\alpha }$, at least one of which is nonempty, and such that every element of L is less than every element of U, there is an element x in F with x larger than every element of L and smaller than every element of U. This is closely related to the model-theoretic property of being a saturated model; any two real closed fields are η_α if and only if they are $\aleph _{\alpha }$-saturated, and moreover two η_α real closed fields both of cardinality $\aleph _{\alpha }$ are order isomorphic.

The generalized continuum hypothesis

The characteristics of real closed fields become much simpler if we are willing to assume the generalized continuum hypothesis. If the continuum hypothesis holds, all real closed fields with cardinality the continuum and having the η₁ property are order isomorphic. This unique field Ϝ can be defined by means of an ultrapower, as ${\mathbb {R}}^{\mathbb {N}}/{\mathbf {M} }$, where M is a maximal ideal not leading to a field order-isomorphic to ${\mathbb {R}}$. This is the most commonly used hyperreal number field in non-standard analysis, and its uniqueness is equivalent to the continuum hypothesis. (Even without the continuum hypothesis we have that if the cardinality of the continuum is $\aleph _{\beta }$ then we have a unique η_β field of size η_β.)

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 ${\mathbb {R}}((G))$ of formal power series on the Sierpiński group.

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 $\aleph _{1}$, Κ has cardinality $\aleph _{2}$, and contains Ϝ as a dense subfield. It is not an ultrapower but it is a hyperreal field, and hence a suitable field for the usages of nonstandard analysis. It can be seen to be the higher-dimensional analogue of the real numbers; with cardinality $\aleph _{2}$ instead of $\aleph _{1}$, cofinality $\aleph _{1}$ instead of $\aleph _{0}$, and weight $\aleph _{1}$ instead of $\aleph _{0}$, and with the η₁ property in place of the η₀ property (which merely means between any two real numbers we can find another).

Examples of real closed fields

• the real algebraic numbers
• the computable numbers
• the definable numbers
• the real numbers
• superreal numbers
• hyperreal numbers
• the Puiseux series with real coefficients

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• Basu, Saugata, Richard Pollack, and Marie-Françoise Roy (2003) "Algorithms in real algebraic geometry" in Algorithms and computation in mathematics. Springer. ISBN 3-540-33098-4 (online version)
• Michael Ben-Or, Dexter Kozen, and John Reif, The complexity of elementary algebra and geometry, Journal of Computer and Systems Sciences 32 (1986), no. 2, pp. 251–264.
• Caviness, B F, and Jeremy R. Johnson, eds. (1998) Quantifier elimination and cylindrical algebraic decomposition. Springer. ISBN 3-211-82794-3
• 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.
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 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
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• Alfred Tarski (1951) A Decision Method for Elementary Algebra and Geometry. Univ. of California Press.
• [Paul Erdos] (1955) P. Erdös, L. Gillman, and M. Henriksen, An isomorphism theorem for real-closed fields, Ann. of Math. (2) 61 (1955), 542–554. MR 0069161 (16,993e)

External links

• Real Algebraic and Analytic Geometry Preprint Server
• Model Theory preprint server