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Topic: Field extension

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Galois extension

Galois group

Dedekind zeta function

Algebraic number

Galois theory

Abelian

Transcendental number

Riemann zeta function

Isomorphism

Map (mathematics)

Phase equilibrium

Evariste Galois

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	PlanetMath: simple field extension
		In this case, we have a finite field extension
		Cross-references: simple infinite field extension, indeterminate, rational functions, simple extension, fields of fractions, implies, isomorphism, homomorphism, finite field extension, algebraic, irreducible polynomial, principal ideal ring, prime ideal, zero divisors, integral domain, subring, polynomials, ideal, residue class ring, isomorphic, ring, image, theorem, ring homomorphism, substitution homomorphism, field
		This is version 19 of simple field extension, born on 2004-05-31, modified 2005-02-15.
	planetmath.org /encyclopedia/SimpleFieldExtension.html (131 words)

	Extension Field -- from Wolfram MathWorld
		For example, the complex numbers are an extension field of the real numbers, and the real numbers are an extension field of the rational numbers.
		The extension field degree (or relative degree, or index) of an extension field
		extension field degree of the extension is the smallest integer
	mathworld.wolfram.com /ExtensionField.html (433 words)

	Field extension - Wikipedia, the free encyclopedia
		Field extensions can be generalized to ring extension which consists of a ring and one of its subrings.
		The field of complex numbers C is an extension field of the field of real numbers R, and R in turn is an extension field of the field of rational numbers Q.
		The significance of Galois extensions and Galois groups is that they allow a complete description of the intermediate fields: there is a bijection between the intermediate fields and the subgroups of the Galois group, described by the fundamental theorem of Galois theory.
	en.wikipedia.org /wiki/Field_extension (1579 words)

	RFC 2774 (rfc2774) - An HTTP Extension Framework
		Extension Declarations An extension declaration can be used to indicate that an extension has been applied to a message and possibly to reserve a part of the header namespace identified by a header field prefix (see 3.1).
		Header field prefixes allow an extension declaration to dynamically reserve a subspace of the header space in a protocol message in order to prevent header field name clashes and to allow multiple declarations using the same extension to be applied to the same message without conflicting.
		The extension definition may be made available in different representations ranging from o a human-readable specification defining the extension semantics (see for example [7]), o downloadable code which implements the semantics defined by the extension, o a formal interface description provided by the extension, to o a machine-readable specification defining the extension semantics.
	www.faqs.org /rfcs/rfc2774.html (4158 words)

	12: Field theory and polynomials
		Algebraic extensions are those in which every element of K is the root of a polynomial with coefficients in F; elements of K which are not algebraic are transcendental over F (e.g.
		Fields of functions of algebraic varieties (essentially the quotient fields of rings F[x1,...,xn]/(P) where P is a multivariable polynomial) are more properly treated in 14: Algebraic Geometry, although these are really just discussions of fields of finite transcendence degree over the ground field.
		Likewise fields of meromorphic functions and local rings of germs of functions are usually treated with their applications to 30: Complex Analysis, 32: Several complex variables, and 58: Analysis on manifolds.
	www.math.niu.edu /~rusin/known-math/index/12-XX.html (1782 words)

	LENS Manual: How To Create Fields in an Object or Extension
		The field is a pointer to an object.
		The field is an array of primitive objects or an array of structures.
		The field is a two-dimensional array of pointers to objects.
	tedlab.mit.edu /~dr/Lens/progExtension.html (905 words)

	Subfields and extension fields
		A subfield of a field is a subset of the field that is also a field.
		In this discussion about extension fields, the extension obtained has been stated to be a field, and seems to obey the properties of a field for the cases examined.
		That the extensions are in fact fields may be rigorously established, but requires some theoretical machinery (regarding maximal ideals) which we are not ready for yet.
	www.engineering.usu.edu /classes/ece/7670/lecture3/node5.html (1081 words)

	Field Ornithology Courses - UCR Extension
		The "Field Study of Birds" classes are led by Eugene Cardiff (pictured on left), retired Curator of Natural History at the San Bernardino County Museum and an ornithologist for more than 35 years.
		Emphasis on identification of breeding plumages in the field and museum.
		A one-week field study of birds in southeastern Arizona, one of the premier locations in the U.S. to observe and study birds.
	www.extension.ucr.edu /sciences/ns/birds/courses.html (1314 words)

	North Carolina Cooperative Extension: Field Crops
		Field crops -- tobacco, soybeans, corn, cotton, peanuts, forages and small grains -- pervade North Carolina's rural landscape.
		North Carolina Cooperative Extension helps by providing growers with the latest research-based information and recommendations on the best varieties and production practices for the state's conditions.
		Extension also helps growers comply with changing regulations, investigate new higher-value opportunities, develop solid business plans and explore marketing options.
	www.ces.ncsu.edu /index.php?page=fieldcrops (232 words)

	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.
		A field is a commutative ring (F, +, *) such that 0 does not equal 1 and all elements of F except 0 have a multiplicative inverse.
		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) (1347 words)

	Paine Field Boulevard Extension - Comments (Site not responding. Last check: 2007-10-20)
		The Consultant and Staff were charged with identifying several alternative alignments for Paine Field Blvd. Extension including analyzing the potential impacts of widening existing SR 525 to accommodate future traffic growth, ferry, local, and regional.
		The Paine Field Blvd. Extension project does not propose ramp connections to Mukilteo Blvd. The technical memorandum was prepared early in the study with projected traffic volumes and levels of service given different scenarios.
		Connecting the Paine Field Blvd. Extension with Mukilteo Blvd. was not carried forward for further analysis because of feedback received very early in the study.
	www.ci.mukilteo.wa.us /area_projects/pfbe/pfbe_comments.htm (7158 words)

	Learning to Operationalise Participatory Processes to Rural Extension
		It describes the demanding profile of extension agents who engage in process facilitation, which is a radical move away from technically based extension towards broader development of rural communities’ capacities to solve problems and to innovate.
		At all levels, and especially at field level, there is a need for a deliberate change in attitudes and behaviour of extension agents and a growth in capabilities to facilitate social processes.
		Most were recruited from the pilot group of 23 field extension agents, and their training skills were further developed through training and coaching by external facilitators/specialists.
	www.iirr.org /PTD/Cases/Moyo.htm (5270 words)

	Field Crops - Graham County (ACIS)
		The Field Crops Program is conducted by Graham County Extension Agent, Randy Norton.
		Field crops are cultivated on approximately 800 thousand acres in Arizona with an estimated value of 617 million dollars.
		Field crop production in the desert southwest is intensely managed with relatively high input cost.
	ag.arizona.edu /crops/counties/graham/fieldcrops.html (136 words)

	Abelian Extension -- from Wolfram MathWorld
		is the field of rational numbers with the square root of two adjoined, a extension field degree-two extension of
		In an Abelian extension that is a splitting field for a polynomial
		Abelian extensions are in a sense the simplest kind of extension because Abelian groups are easier to understand than more general ones.
	mathworld.wolfram.com /AbelianExtension.html (225 words)

	New Ohio State Extension Crop Field Guide Available
		The Corn, Soybean, Wheat and Alfalfa Field Guide, developed by researchers and Extension agents, is designed to serve as a quick reference handbook for growers, scouts and consultants who have questions about crop condition or performance problems while scouting fields.
		The guide is a hands-on reference for growers to use out in the field when trying to assess field problems," said Peter Thomison, an Ohio State Extension agronomist and one of 15 authors of the publication.
		The 240-page field guide is packed full of information concerning Ohio's four major field crops: corn, soybeans, wheat and alfalfa.
	www.ag.ohio-state.edu /~news/story.php?id=2153 (311 words)

	Finite Fields
		Any extension field L of K can be viewed as a vector space over K and the dimension of this vector space is called the degree of L over K, and is denoted by [L:K].
		Furthermore, L is isomorphic to the quotient field K[x]/, where denotes the principal ideal of K[x] generated by f(x).
		We denote the group of all automorphisms of a field L by G(L) and the subgroup of G(L) that fixes all elements of the subfield K of L by G(L/K).
	www-math.cudenver.edu /~wcherowi/courses/finflds.html (3085 words)

	Algebraic extension - Wikipedia, the free encyclopedia
		In abstract algebra, a field extension L /K is called algebraic if every element of L is algebraic over K, i.e.
		For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.
		Every field has an algebraic extension which is algebraically closed (called its algebraic closure), but proving this in general requires some form of the axiom of choice.
	en.wikipedia.org /wiki/Algebraic_extension (345 words)

	RIGZONE - South Alibek-2 Well Confirms Northern Extension of Field
		This crestal well confirms the field structure and indicates the presence of the reservoir zones significantly north of previous wells.
		The Company estimates that approximately 1250 acres has been delineated by the three new wells, over an area of the field that is three miles in length and half a mile in width.
		The Company estimates the field could have an areal extent exceeding 10,000 acres based on the available seismic data, most of which still remains to be confirmed by future wells.
	www.rigzone.com /news/article.asp?a_id=10712 (541 words)

	ArcPad Extension Enhances Field Productivity
		The ArcPad StreetMap extension provides users with ready-to-use, quality data for the United States with predefined symbology, advanced address matching, geocoding, and routing capabilities.
		Although the ability to calculate routes in the field can be invaluable, sometimes generating on-the-fly stop points or barriers is impossible, unnecessary, or undesirable.
		Although GRF documents are currently generated and maintained only by the ArcPad StreetMap extension, a simple custom ArcMap script—available from the ESRI Web site—allows users to select features and write the necessary information to a GRF document.
	www.esri.com /news/arcuser/0403/ap_extension.html (1047 words)

	Extension Daily: Extension-Sponsored Field Day Focuses on Beef Quality and Safety
		The producer was in east Alabama Oct. 27 to attend the regional Beef Cattle Field Day, sponsored by the Alabama Cooperative Extension System offices in Lee and Chambers counties.
		Ever the optimist, Bobby “Smokey” Spears, a retired Extension agent who organized the field day along with Chambers County Extension Coordinator Kim Wilkins, opened the training by expressing gratitude for the rain and confidence that the field day would prove successful regardless of the inclement weather.
		The field day focused on four topics of vital interest to beef producers: herd health issues, premise identification, electronic ear tagging and body confirmation and conditioning.
	www.aces.edu /department/extcomm/npa/daily/archives/002490.php (510 words)

	Field Development Geosciences
		The key to understanding the performance and potential of a field is grasping the geologic framework and petroleum system.
		Behind pipe, field extension, and deeper potential reservoirs will be identified.
		The delineation of the geometry of a field's reservoirs is key to efficient development.
	www.fielddevelopment.com /GeoSciences.htm (296 words)

	Extension Service has new frost damage Web site
		The University of Minnesota Extension Service is responding to the recent frosts and unusually cold growing season with a new Web site, http://www.extension.umn.edu/frostdamage.
		New articles by Extension experts are included on harvesting, hay and silage, crop insurance, drying and storage.
		Members of the College of Agricultural, Food and Environmental Sciences and field based Extension educators are key members of the Frost Response Team.
	www.extension.umn.edu /extensionnews/2004/frost.html (321 words)

	Simple, Low-Cost Data Collection Methods for Agricultural Field Studies (Site not responding. Last check: 2007-10-20)
		Because several states also have granted Extension agents faculty status, the requirement for refereed publications as evidence of scholarly activity is now becoming more common for field agents as well.
		Loveridge (1998) noted that Extension field personnel can be at a disadvantage when conducting research because of the distance from a university campus and lack of infrastructure (equipment) and monetary support.
		Techniques for field evaluation of alfalfa quality parameters (primarily acid and neutral detergent fibers) based on the tallest stem and maturity of the most mature stem have been developed in several states (Owens, Albrecht, and Hintz, 1995; Sulc et al., 1997).
	www.joe.org /joe/2000april/a1.html (1993 words)

	RFC 3608 - Session Initiation Protocol (SIP) Extension Header Field for Service Route Discovery During Registration
		This new header field indicates a "preloaded route" that the UA may wish to use if requesting services from the proxy network associated with the registrar generating the response.
		The mechanism by which the registrar constructs the header field value is specific to the local implementation and outside the scope of this document.
		The allowable usage of header fields is described in Tables 2 and 3 of [3].
	www.packetizer.com /rfc/rfc.cgi?num=3608 (3897 words)

	A Graphical Field Extension for sky
		As the instrument field is dragged with the middle mouse button, its right ascension and declination are displayed next to the cross hair that moves along with it.
		In summary, the middle mouse button is used to drag the field, the right mouse button is used to interrogate the field, and the left mouse button is reserved for selecting stars, selecting pointing origins, and button clicks.
		Moreover, any field rotation that takes place during the exposure will be centered about these two coincident points.
	www.cv.nrao.edu /adass/adassVI/conrada.html (1527 words)

	Field Investigation Unit - PSU Veterinary Extension
		The field studies unit is an integral part of both the Department of Veterinary Science, PSU, and the Pennsylvania Bureau of Animal Health and Diagnostics.
		Problem-solving: Veterinarians from the field studies group respond to requests for assistance from livestock and poultry producers, veterinarians, county agents, and other agricultural resources.
		Results of directed field studies are communicated to producers and the scientific and ag business community through farm and professional journals as well as news releases and presentations.
	vetextension.psu.edu /fi_farm.asp (368 words)

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