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Topic: Ordered field

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Field (mathematics)

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Real number

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	Field (mathematics) - Wikipedia, the free encyclopedia
		In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
		Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the rational numbers, real numbers, and complex numbers.
		For a given field F, the set F(X) of rational functions in the variable X with coefficients in F is a field; this is defined as the set of quotients of polynomials with coefficients in F.
	en.wikipedia.org /wiki/Field_(mathematics) (1581 words)

	Ordered field: Definition and Links by Encyclopedian.com - All about Ordered field (Site not responding. Last check: 2007-11-07)
		The smallest subfield is isomorphic to the rationals (as for any field of characteristic 0), and the order on this rational subfield is the same as the order of the rationals themselves.
		If F is equipped with the order topology arising from the total order <=, then the axioms guarantee that the operations + and * are continuous.
		Finite fields cannot be turned into ordered fields, because they do not have characteristic 0.
	www.encyclopedian.com /or/Ordered-field.html (354 words)

	Learn more about Real number in the online encyclopedia. (Site not responding. Last check: 2007-11-07)
		However, an ordered 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 completeness (topology); the description in the section Completeness above is a special case.
		By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical.
		Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and thus are not Archimedean.
	www.onlineencyclopedia.org /r/re/real_number.html (2010 words)

	Real closed field - Wikipedia, the free encyclopedia
		If F is any ordered field, 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 ordering on F, and is unique up to order isomorphism.
		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.
		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 hyperreal numbers is real closed and non-archimedean.
	en.wikipedia.org /wiki/Real_closed_field (866 words)

	Ordered Fields (Site not responding. Last check: 2007-11-07)
		A field is "ordered" if its nonzero elements can be split into two sets, p and n, such that p contains x iff n contains -x, and p is closed under addition and multiplication.
		The elements of the field are linearly ordered.
		The rationals and the reals are ordered, but complex numbers cannot be, since neither i nor -i can be positive, their squares being -1.
	www.mathreference.com /fld,ordered.html (335 words)

	Hyperreal number - Open Encyclopedia (Site not responding. Last check: 2007-11-07)
		Being a hyperreal field is a stronger condition than being a real closed field strictly containing $\Bbb{R}, and also stronger than being a superreal$ field in the sense of Dales and Woodin.
		The hyperreal fields we obtain in this case are called ultrapowers of $\Bbb{R} and are identical to the ultrapowers constructed via free ultrafilters in model theory.$
		The map st is locally constant, which entails that its derivative is identically zero and that it is continuous with respect to the order topology on the finite hyperreals.
	open-encyclopedia.com /Hyperreal_number (1814 words)

	Semiannual Variation of Geomagnetic Activity
		In order not to obscure the basic mechanism in our model, -we have made two simplifying assumptions: first, that the interplanetary field lies along the Parker spiral angle; and, second, that the field is toward the sun as often as it is away from the sun.
		Since the spring peak is due to fields pointing toward the sun and the fall peak is due to fields pointing away from the sun, the first polarity amplifies the spring and fall peaks and the second polarity reduces the peaks.
		A model, in which the interaction, ordered in solar magnetospheric coordinates, is zero for northward components of the interplanetary field while the interaction is proportional to magnitude of the southward components predicts the correct phase and provides a yearly variation in the strength of the interaction sufficient to cause the observed effect.
	www-ssc.igpp.ucla.edu /personnel/russell/papers/40 (8728 words)

	PlanetMath: every ordered field with the least upper bound property is isomorphic to the real numbers (Site not responding. Last check: 2007-11-07)
		"every ordered field with the least upper bound property is isomorphic to the real numbers" is owned by archibal.
		proof that every ordered field with the least upper bound property is isomorphic to
		This is version 1 of every ordered field with the least upper bound property is isomorphic to the real numbers, born on 2004-02-20.
	planetmath.org /encyclopedia/EveryOrderedFieldWithTheLeastUpperBoundPropertyIsIsomorphicToTheRealNumbers.html (142 words)

	Real number - FreeEncyclopedia (Site not responding. Last check: 2007-11-07)
		Construction by Dedekind cuts -- A Dedekind cut in an ordered field is a partition of it, (A, B), such that A is nonempty and closed downwards, B is nonempty and closed upwards, and A has no maximum.
		Ordered fields that satisfy the same first-order sentences as R are called nonstandard models[?] of R.
		Self-adjoint operators 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.
	openproxy.ath.cx /re/Real_numbers.html (2236 words)

	Ordered field -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		The smallest subfield is isomorphic to the (Click link for more info and facts about rationals) rationals (as for any field of characteristic 0), and the order on this rational subfield is the same as the order of the rationals themselves.
		If every element of an ordered field lies between two elements of its rational subfield, then the field is said to be (Click link for more info and facts about Archimedean) Archimedean.
		If F is equipped with the order (The configuration of a communication network) topology arising from the total order ≤, then the axioms guarantee that the operations + and * are (Click link for more info and facts about continuous) continuous.
	www.absoluteastronomy.com /encyclopedia/o/or/ordered_field.htm (613 words)

	Hyperreal number (Site not responding. Last check: 2007-11-07)
		In mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as R) denote an ordered* field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle.
		As a real closed field with cardinality the continuum, it is isomorphic as a field to R but is not isomorphic as an ordered field to R.
		The hyperreal fields we obtain in this case are called ultrapowers of R and are identical to the ultrapowers constructed via free ultrafilters in model theory.
	www.worldhistory.com /wiki/H/Hyperreal-number.htm (2146 words)

	Analysis WebNotes: Chapter 02, Class 03
		As long as the axioms of the linear order and the arithmetic operations are independent, we can't expect to be able to deduce any connection.
		{0,1} is not an ordered field with the order 0< 1.
		As we remarked above, all the algebraic properties of inequalities that we need can be deduced from the axioms for an ordered field (usually from a combination of the rules for field, for an ordered set, and for the two additional axioms for ordered fields).
	www.math.unl.edu /~webnotes/classes/class03/class03.htm (265 words)

	Fourth Annual COLLOQUIUMFEST - Titles and abstracts of talks
		Hardy fields and fields of transseries are H-fields.
		An optimistic conjecture is that each H-field can be embedded, as ordered differential field, into a field of transseries, analogous to the fact that each valued field of equal characteristic zero can be embedded as valued field into a field of generalized power series.
		Let $R$ be an ordered structure and let the open core $R^{\circ}$ of $R$ be the reduct of this structure whose basic relations are the open definable sets in $R$ in any number of variables.
	math.usask.ca /fvk/mb4t.htm (648 words)

	PlanetMath: ordered ring (Site not responding. Last check: 2007-11-07)
		Examples of rings which do not admit any ordering relation making them into an ordered ring include:
		Cross-references: prime, finite field, complex numbers, ordering relation, examples of rings, relation, Variable, polynomial ring, real numbers, ordering, integers, field, total ordering, commutative ring
		This is version 7 of ordered ring, born on 2001-10-21, modified 2004-10-24.
	planetmath.org /encyclopedia/OrderedField.html (96 words)

	Garden Watchdog: The scoop on 'Henry Field's Seed & Nursery'
		Ordered a nectarine, plum, and seeds, with a $20 off $40 coupon, got the seeds almost immediately, the trees very early for zone 5, they shipped a differnet nectarine due to being out of the one I wanted.
		The strawberries I ordered were supposed to be runnerless, but within no time at all they escaped the pot I had them in and took over a patch of the garden.
		Every company I have ever ordered seed from (with the exception of Pinetree) has had at least one item back-ordered, I've learned to order early enough to be able to place another order with another company before the growing season; so I don't consider one item to be a negative.
	davesgarden.com /gwd/c/185 (5349 words)

	SPACE.com -- Galaxy Makes Order out of Chaos
		The Milky Way has a smooth magnetic field that scientists have assumed was created over billions of years by the relatively undisturbed rotation of the galaxy and a fairly smooth distribution of stellar birth and death.
		Yet the LMC's magnetic field is not at all disorganized, based on the most comprehensive map ever made of another galaxy's magnetism.
		Astrosociology is the study of astrosocial phenomena (social/cultural patterns related to space), a multidisciplinary field open to all scientists and engineers interested in "space and society" issues.
	www.space.com /scienceastronomy/050315_galaxy_chaos.html (703 words)

	Articles - Real number (Site not responding. Last check: 2007-11-07)
		The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
		The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:
		It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element z, z + 1 is larger), so this is not the sense that is meant.
	www.gaple.com /articles/Real_number (1753 words)

	What are the 'real numbers,' really?
		The reals and the rationals, with their usual orderings are two familiar examples of ordered fields.
		The ordered fields that have no infinitesimals are called Archimedean fields; we'll see later that the real number system (i.e., Dedekind's number system, also known as the standard real numbers) is Archimedean.
		Its ordering is the same as the ordering of the set of rational functions.
	www.math.vanderbilt.edu /~schectex/courses/thereals (5085 words)

	Official Records : Page 548 \| THE PENINSULAR CAMPAIGN,VA. Chapter XXIII. (Site not responding. Last check: 2007-11-07)
		On the 27th I was ordered to engage a battery of the enemy near the house of Dr. Garnett.
		I took position in the open field in front of the enemy's battery, which was protected by formidable earthworks.
		On the 28th I was ordered, with a section of my battery, to take position near the Chickahominy on the New Bridge road, and open on the enemy near Mr.
	ehistory.osu.edu /uscw/library/or/013/0548.cfm (402 words)

	Ordering
		Given a ring, we can say that it is ordered when you have a special subset of that ring behaves in a very special way.
		Now, we gave the definition for an ordered ring, we can also define an ordered field the same way.
		There is another definition which is probably just as good, but may or may not be easier: A field is algebraically closed if every polynomial splits into linear factors.
	db.uwaterloo.ca /~alopez-o/math-faq/node10.html (784 words)

	Ordered Field Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-11-07)
		Looking For ordered field - Find ordered field and more at Lycos Search.
		Find ordered field - Your relevant result is a click away!
		Look for ordered field - Find ordered field at one of the best sites the Internet has to offer!
	www.karr.net /search/encyclopedia/Ordered_field (544 words)

	[KKSh:601] (Site not responding. Last check: 2007-11-07)
		an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a non-surjective logarithm.
		For an arbitrary ordered field k, no exponential on k((G)) is compatible, that is, induces an exponential on k through the residue map.
		This is proved by showing that certain functional equations for lexicographic powers of ordered sets are not solvable.
	www.math.rutgers.edu /pub/shelah/all/601_abs.html (92 words)

	COURT ORDERED FEDERAL EMPLOYEES HEALTH BENEFIT INDICATOR (Site not responding. Last check: 2007-11-07)
		The Court Ordered FEHB Indicator restricts FEHB changes for an employee under a court or administrative order to provide health benefits coverage.
		Although EEX will allow an employee under court order to change to another family plan, additional processing and authorization are required.
		Information for mandatory FEHB coverage may be included in a child support withholding order or other administrative order.
	www.va.gov /publ/direc/finance/04ga2_04bul.htm (602 words)

	CCAPrint Newsletter for Model 204 and System 1032 Users - December 2002
		Fields most often go missing during file loads or file reorganizations due to errors in the data being loaded or errors in the logic of the load program itself.
		In Figures 1 through 4, the field RECTYPE has only two values, A and B, so the number of Table D Ordered Index pages, which must be scanned looking for those values and associated record lists or bit patterns, is naturally small.
		If you need to know whether a particular field is or is not present in a large set of records, define that field as ORDERED CHARACTER and use the facilities of the Ordered Index as illustrated.
	www.cca-int.com /resources/ccaprint/news0212.html (1222 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		\item{} (b) Deduce that there is a unique total order for the field $Q$ of fractions of $R$ extending the given ordering for $R$.
		We say that $R$ (or the ordering $\le$) is \ut{Archimedean} in case for all $a, b >0$ there exists $n \in \fatn$ with $na>b$.
		Assuming the lexicographic ordering, find necessary and sufficient conditions for $f/g$ to be infinitely small (infinitely large).
	darkwing.uoregon.edu /~anderson/math649/lecture26.html (983 words)

	Ordered field - Wikipedia
		the surreal numbers; in fact every ordered field can be embedded into the surreal numbers
		Finite fields and the complex numbers cannot be turned into ordered fields.
		This page was last modified 20:01, 17 August 2001.
	nostalgia.wikipedia.org /wiki/Ordered_field (249 words)

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