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Topic: Real closed field

	Real number - Open Encyclopedia (Site not responding. Last check: 2007-10-09)
		Real numbers may be rational or irrational; algebraic or transcendental; and positive, negative, or zero.
		The nonexistence of a subset of the reals with cardinality strictly in between that of the integers and the reals is known as the continuum hypothesis.
		The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalised such that the unit interval [0,1] has measure 1.
	open-encyclopedia.com /Real_number (1982 words)

	Real closed field -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-09)
		A crucially important property of the real numbers is that it is an (additional info and facts about archimedean field) archimedean field, meaning it has the archimedean property that for any real number, there is an integer larger than it in absolute value.
		A non-archimedean field is, of course, a field that is not archimedean, and there are real closed non-archimedean fields; for example any field of (additional info and facts about hyperreal numbers) hyperreal numbers is real closed and non-archimedean.
		This unique field Ϝ can be defined by means of an (additional info and facts about ultrapower) ultrapower, as, where M is a maximal ideal not leading to a field order-isomorphic to.
	www.absoluteastronomy.com /encyclopedia/r/re/real_closed_field.htm (947 words)

	Real number - Metaweb (Site not responding. Last check: 2007-10-09)
		The nonexistence of a subset of the reals with cardinality strictly in between that of the integers and the reals is known as the continuum hypothesis (http://en.wikipedia.org/wiki/continuum_hypothesis).
		The reals are a contractible (http://en.wikipedia.org/wiki/contractible) (hence connected (http://en.wikipedia.org/wiki/connected) and simply connected (http://en.wikipedia.org/wiki/simply_connected)), locally compact (http://en.wikipedia.org/wiki/local_compactness) separable (http://en.wikipedia.org/wiki/separable) metric space, of dimension (http://en.wikipedia.org/wiki/dimension) 1, and are everywhere dense (http://en.wikipedia.org/wiki/first_category).
		The reals carry a canonical measure (http://en.wikipedia.org/wiki/measure), the Lebesgue measure (http://en.wikipedia.org/wiki/Lebesgue_measure), which is the Haar measure (http://en.wikipedia.org/wiki/Haar_measure) on their structure as a topological group (http://en.wikipedia.org/wiki/topological_group) normalised such that the unit interval (http://en.wikipedia.org/wiki/unit_interval) [0,1] has measure 1.
	www.metaweb.com /wiki/wiki.phtml?title=Real_number&printable=yes (1822 words)

	Learn more about Real number in the online encyclopedia. (Site not responding. Last check: 2007-10-09)
		Real numbers could be constructed as the topological completion of rational numbers.
		Occasionally, formal elements +∞ and -∞ are added to the reals to form the extended real number line, a compact space which is not a field anymore but retains many of the properties of the real numbers.
		Self-adjoint operatorss on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra.
	www.onlineencyclopedia.org /r/re/real_number.html (2010 words)

	Real number : Real
		The real numbers are, intuitively, numbers that are in one-to-one correspondence with the points on a line -- the number line[?].
		Real numbers may be expressed by decimal fractions, such as 324.823211247...; it is recursive[?] if the digits can be specified by a recursive algorithm.
		Occasionally, formal elements +∞ and -∞ are added to the reals to form the extended real number line, a compact space which isn't a field anymore but retains many of the properties of the real numbers.
	www.wordlookup.net /re/real.html (2476 words)

	Algebraically closed field
		By contrast, the field of complex numbers is algebraically closed, which is the content of the fundamental theorem of algebra.
		Every field F has an "algebraic closure", which is the smallest algebraically closed field of which F is a subfield.
		In particular, the field of complex numbers is the algebraic closure of the field of real numbers.
	www.teachersparadise.com /ency/en/wikipedia/a/al/algebraically_closed_field.html (188 words)

	PlanetMath: real closed fields
		theory is the theory of the real numbers.
		The semi algebraic sets on a real closed field are boolean combinations of solution sets of polynomial equalities and inequalities.
		This is version 3 of real closed fields, born on 2003-01-23, modified 2005-04-05.
	planetmath.org /encyclopedia/RealClosedFields.html (289 words)

	PlanetMath: infinitesimal
		be a real closed field, for example the reals thought of as a structure in
		is the intersection of the reals with the algebraic numbers.
		This is due to Abraham Robinson, who used such fields to formulate nonstandard analysis.
	planetmath.org /encyclopedia/Infinitesimal2.html (217 words)

	Glossary of field theory (Site not responding. Last check: 2007-10-09)
		Field theory is the branch of mathematics in which fields are studied.
		Subfield : A subfield of a field F is a subset of F which is closed under the field operation + and * of F and which, with these operations, forms itself a field.
		Algebraically closed field : A maximal algebraic extension field of F is its algebraic closure.
	www.worldhistory.com /wiki/G/Glossary-of-field-theory.htm (887 words)

	Algebraically closed field explained (Site not responding. Last check: 2007-10-09)
		By contrast, the field of complex numbers is algebraically closed: this is stated by the fundamental theorem of algebra.
		In particular, the field of complex numbers is an algebraic closure of the field of real numbers.
		Also, the field of algebraic numbers is the algebraic closure of the field of rational numbers.
	www.wordspider.net /al/algebraically-closed-field.html (513 words)

	Abstracts for Invited Talks at the RSME-AMS Special Session on Effective Analytic Geometry Over Complete Fields
		In analogy with the real Nullenstellensatz, which follows from Artin's representation of positive definite rational functions over real fields as sums of squares (Hilbert's 17th Problem), we describe a p-adic Nullstellensatz that is a consequence of a representation theorem for integral definite rational functions over p-adic fields.
		We describe a quantifier elimination theorem for certain valued fields (including Laurent series fields of characteristic zero and the field of fractions of the Witt vectors of an algebraically closed field of positive characteristic) given together with analytic functions and an endomorphism.
		Hardy fields and fields of transseries are H-fields.
	www.math.tamu.edu /~rojas/sevabs.html (922 words)

	Artin, Emil (Site not responding. Last check: 2007-10-09)
		In his doctoral thesis of 1921 he formulated the analogue of the Riemann hypothesis about the zeros of the classical zeta function, studying the quadratic extension of the field of rational functions of one variable over finite constant fields, by applying the arithmetical and analytical theory of quadratic numbers over the field of natural numbers.
		A year later, in collaboration with Schrier, he succeeded in treating real algebra in an abstract manner, defining a field as real-closed if it itself was real but none of its algebraic extensions were.
		He was then able to demonstrate that a real-closed field could be ordered in an exact manner and that, in it, typical laws of algebra were valid.
	www.cartage.org.lb /en/themes/Biographies/MainBiographies/A/Artin/1.html (332 words)

	Real and Complex Numbers
		Antoine Arnauld, a close friend of Pascal, questioned that -1:1 = 1:-1, because, he said, -1 is less than +1; hence how could the smaller be to the greater as the greater to the smaller [6, page 115,].
		A field is a model for the axioms of field theory.
		To axiomatize the real closed fields, we add in more axioms stating that every positive element has a square root and every polynomial of odd degree has a root.
	www.math.psu.edu /melvin/logic/node8.html (999 words)

	[No title]
		The changes in the opened and closed fields (red and fl lines in the lower panels, respectively) concentrate around the active region longitude, at the edges of the helmet streamer belt.
		We note that in all cases the time elapsed in the sequences shown is arbitrary, The real coronal and interplanetary responses will greatly depend on the rapidity of the changes, and the ability (or inability) of the corona and solar wind to accommodate them in quasistatic fashion.
		It is important to note that some of the changing fields are connected to the strong fields of the bipoles, which simulate active regions or their remains.
	sprg.ssl.berkeley.edu /~jgluhman/cme/gedankengrl (1465 words)

	Godel - Completeness of axiomatic systems - Page 2 - Physics Help and Math Help - Physics Forums
		Real closed field theory only lets you make statements about polynomial equations, not the differential equations of physics (hence my post above).
		I think that the complex numbers, quaternions, octonions, and so on, are merely defined as extensions of the real numbers, with additional axioms which define the i, j, k and the way of doing additions and multiplication of complex numbers or quaternions.
		If the real numbers m and n are not both zero, then there exist real numbers p and q such that (m * p - n * q) = 1 and (m * q + n * p) = 0.
	www.physicsforums.com /showthread.php?p=270075 (1653 words)

	Atlas: A Nullstellensatz for Henselian fields with real-closed residue field by Rafel Farre (Site not responding. Last check: 2007-10-09)
		A Nullstellensatz for Henselian fields with real-closed residue field
		This result extends all (known to us) previous results for such kind of fields, in particular the well-known results of Krivine-Dubois-Rissler-Stengle (real-closed fields), Jacob [J] (R((t)) and real-series closed fields) and Becker-Jacob [B-J] (generalized real-closed fields).
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacv-58.
	atlas-conferences.com /cgi-bin/abstract/cacv-58 (226 words)

	Re: Specifying the Reals in LSL? (Site not responding. Last check: 2007-10-09)
		Defining the reals in terms of Cauchy sequences is akin to implementing an abstract data type.
		The definition "works" because Cauchy sequences satisfy the properties we expect of the reals, but we want to use those properties (and not an exposed representation) when reasoning about the reals.
		(A polynomial trait would come in handy here.) Now you've got the axioms for what is known as a real closed field, and you are almost home, because Tarski showed that the first-order theory of the reals is equivalent to the theory of real closed fields of characteristic zero.
	nms.lcs.mit.edu /spd/larch/archive/msg00300.html (248 words)

	Physics Help and Math Help - Physics Forums - Godel - Completeness of axiomatic systems
		I note that in the article on the reals there is a lot about their completeness, but this refers to a completely different concept - the convergence of Cauchy Sequences.
		The (topological) completeness of the "real numbers" does depend on the "sets" you're allowed to use; the algebraic numbers, real numbers, and hyperreal numbers all satisfy the axioms of the theory of real numbers...
		Of course I could simply add to the real closed field axioms one saying that for every number x there is a number known as sin(x).
	www.physicsforums.com /printthread.php?t=36204 (1411 words)

	ORDINAL REAL NUMBERS 2
		In this topology, as it is known, the field F is a topological field.
		As in the case of fields that are classes, we may permit topological spaces that are classes and the open sets is a class of subclasses closed to union and finite intersection.
		In other words the fields of ordinal real numbers are Archemidean complete fields (although they may be non-Archemidean).But this is a characteristic property of the fields of transfinite real numbersb of Glayzal.
	www.softlab.ntua.gr /~kyritsis/PapersInMaths/InfinityandStochastics/OR2.htm (2766 words)

	Algebraically closed field : Algebraically closed
		A field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F.
		If one adjoins to F all roots of all polynomials, the resulting field is called the algebraic closure of F.
		For example, C (the field of complex numbers) is the algebraic closure of R (the field of real numbers).
	www.fastload.org /al/Algebraically_closed.html (194 words)

	Fourth Annual COLLOQUIUMFEST - Titles and abstracts of talks
		It is well known that many results of real semialgebraic geometry generalize to any real closed field.
		Specifically, we show that these fields fit into the context of rosy theories, and are, thus, equipped with a good independence theory and dimension theory.
		I will outline a geometric proof of the following well-known theorem for o-minimal expansions of the real field: the Hausdorff limits of a compact, definable family of sets are definable.
	math.usask.ca /fvk/mb4t.htm (648 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Subject: algebraic topology and the real closed field Let R be a real closed field and R[i] is its algebraic closure.
		It is known that all properties of the real field and the complex field also holds for R and R[i] respectively except the completeness.
		The results (i) and (ii) are used by Wood (1985) to prove the existence of singular eigenvalue of quaternion matrices.
	www.lehigh.edu /~dmd1/real514 (127 words)

	Real number - Metaweb (Site not responding. Last check: 2007-10-09)
		in flboard bold)" to represent the set of all real numbers.
		Axiomatic approach Let R denote the set of all real numbers.
		However, an ordered (mathematics) group (and a field is a group under the operations of addition and subtraction) defines a uniform structure, and uniform structures have a notion of topological completeness; the description in the section Completeness above is a special case.
	www.metaweb.com /wiki/wiki.phtml?title=Real_number&printable=yes (1783 words)

	ORDINAL REAL NUMBERS 1
		This is the third paper of a series of five papers that have as goal the definition of topological complete linearly ordered fields (continuous numbers) that include the real numbers and are obtained from the ordinal numbers in a method analogous to the way that Cauchy derived the real numbers from the natural numbers.
		The relevancy with the surreal numbers and the non-standard (hyper) real numbers,shall be studied in a later paper.
		It is elementary in algebra that if the integral domain is linearly ordered then also its field of quotients (localization) with the previous definition for its set of positive elements, is a linearly ordered field with the restriction of its ordering on the integral domain to coincide with the ordering of the integral domain.
	www.softlab.ntua.gr /~kyritsis/PapersInMaths/InfinityandStochastics/OR1.htm (3795 words)

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