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Topic: Characteristic (field)

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	PlanetMath: Q is the prime subfield of any field of characteristic 0, proof that
		This is version 13 of proof that Q is the prime subfield of any field of characteristic 0, born on 2006-02-06, modified 2006-03-13.
		The formal counterpart to what you're talking about is the prime field, and your two theorems refer to those, Q and F_p (which, incidentally, is a better notation since Z_p can be confused with the p-adic integers).
		I recommend either giving a formal definition of a prime field, or re-write to be clear that "ground field" is not a formal term, and that the expression is used informally by mathematicians to refer to an obvious choice of an important field lying around somewhere.
	planetmath.org /encyclopedia/GroundField.html (446 words)

	Glossary of field theory - Wikipedia, the free encyclopedia
		Field theory is the branch of mathematics in which fields are studied.
		The characteristic of the field F is the smallest positive integer n such that n Â·1 = 0; here n Â·1 stands for n summands 1 + 1 + 1 +...
		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.
	en.wikipedia.org /wiki/Glossary_of_field_theory (765 words)

	PlanetMath: characteristic
		Note that the first definition also applies to arbitrary rings, and not just to fields.
		The characteristic of a field (or more generally an integral domain) is always prime.
		This is version 12 of characteristic, born on 2002-01-01, modified 2006-11-18.
	planetmath.org /encyclopedia/Characteristic.html (113 words)

	Finite field (Site not responding. Last check: 2007-10-16)
		Since every field of characteristic 0 contains the rationals and is therefore infinite, all finite fields have prime characteristic.
		The multiplicative group of every finite field is cyclic, a special case of a theorem mentioned in the article about fields.
		Finite fields also find applications in coding theory : many codes are constructed as subspace s of vector space s over finite fields.
	www.serebella.com /encyclopedia/article-Finite_field.html (1192 words)

	FiniteFields
		A field is an algebraic structure obeying the rules of ordinary arithmetic.
		Since it is possible to have fields of the same size using different irreducible polynomials, it is useful to be able distinguish elements from these fields.
		-tuple is assumed to represent a primitive element of the field.
	documents.wolfram.com /v4/AddOns/StandardPackages/Algebra/FiniteFields.html (1268 words)

	Patent 4739513: Method and apparatus for measuring and correcting acoustic characteristic in sound field
		The acoustic transmission characteristics of a sound field, such as in the passenger compartment of a vehicle, are measured by placing microphones at the positions of ears of a dummy mannequin having a sound absorbing characteristic similar to that of a clothed adult human.
		In order to measure the acoustic transmission characteristic in a given sound field, heretofore a method has generally been employed whereby a dummy head on the ears of which microphones are arranged is set at the listening point, and the characteristic is determined from the outputs of the microphones.
		Furthermore, since the characteristic is measured according to the sum of the output signals of the microphones at the ears of the dummy head, the measurement result is a composite one including the directivity of the ears and the summing effect.
	www.freepatentsonline.com /4739513.html (6653 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-16)
		Basic problems in the theory of fields consist of giving a description of all subfields of a given field, of all fields containing a given field, i.e.
		Field theory originated (within the framework of the theory of algebraic equations) in the middle of the 19th century.
		The German term for "field" is "Körper" and this is of course the term used in [a2].
	eom.springer.de /f/f040090.htm (480 words)

	Kummer theory - Wikipedia, the free encyclopedia
		In mathematics, a Kummer extension of fields is a field extension
		Kummer theory is basic, for example, in class field theory and in general in understanding abelian extensions ; it says that in the presence of enough roots of unity, cyclic extensions can be understood in terms of extracting roots.
		The main burden in class field theory is to dispense with extra roots of unity ('descending' back to smaller fields); which is something much more serious.
	en.wikipedia.org /wiki/Kummer_theory (427 words)

	Geometry Seminar Lecture Notes 1
		The set of sums of 1 forms a ring contained in F, and the quotient field of that ring is the smallest field contained in F (a subfield of F).
		For fields of characteristic 0 this smallest subfield is isomorphic to Q, the field of rational numbers, while for fields of characteristic p it is isomorphic to GF(p).
		An automorphism of a field is a bijection of the field onto itself which is both an additive and multiplicative homomorphism (i.e., preserves both addition and multiplication).
	www-math.cudenver.edu /~wcherowi/geom/gsln1.html (1161 words)

	Matrix Reference Manual: Algeraic Structures
		The characteristic of an integral domain or field is always either 0 or a prime number.
		A field with characteristic 0 is infinite but not all infinite fields have 0 characteristic.
		A field with a finite number of elements, k, is a Galois Field and denoted by GF(k).
	www.ee.uwa.edu.au /~roberto/teach/matrix/vector.html (880 words)

	Characteristic (Site not responding. Last check: 2007-10-16)
		In abstract algebra, the characteristic of a ring R is defined to be the smallest positive integer n such that 1R+.
		In abstract algebra, the characteristic of a ring R is defined to be the smallest positive integer n such that 1
		Characteristic is also sometimes used as a piece of jargon in discussions of universals in metaphysics, often in the phrase 'distinguishing characteristics'.
	www.termsdefined.net /ch/characteristic.html (471 words)

	Patent 4051458: Video amplitude related measurements in image analysis
		A second field is used to set the most significant bit of the word by generating a word with the A/D word generator 1819, with most significant bit of the word set to "one" followed by "zeroes".
		During a third field, a word is generated with its next most signficant bit set to "one", which is bit number 7 in a 9-bit word, where bit number 8 is the most significant bit and bit number "zero" is the least significant bit.
		In fields 11 through 26, which are odd-numbered fields, a word is added corresponding to one least significant bit, if the word in memory is to be increased; and "zero" is added if the word is to be decreased.
	www.freepatentsonline.com /4051458.html (11663 words)

	Intelligence and Logos, Chapter 2
		The field thus constituted is so, if I may be permitted the expression, in a private way, because the totality of this field in its three zones (first level, background, periphery) is surrounded at the same time by a line which positively determines what the field encompasses; this is precisely its horizon.
		The field of which we have been speaking can be described first of all through its content, by the things that are in it: rocks, trees, the sea, etc. But the field can and ought to be described according to its own unity.
		The field as the first plane, as the periphery, as the horizon, is just the structure of positionality; i.e., the structure of the "among" as a "toward".
	www.zubiri.org /works/englishworks/si/SI2C2.htm (6114 words)

	Descriptive and Meaningful Database Field Names \| Database Solutions for Microsoft Access \| databasedev.co.uk
		If a field name is ambiguous, unclear or is vague it may suggest that the purpose of the field is not fully identified.
		Fields named "GST_RT" could be extremely misleading, and the user may find these hard to determine the characteristics of the field.
		A field name should be singular as it should represent a single field characteristic of the subject of the table to which it belongs.
	www.databasedev.co.uk /database_field_names.html (685 words)

	Ultrasound (EHC 22, 1982)
		The directivity of the beam in the far field is determined by diffraction, in the same way that a light wave is affected by a small aperture; the higher the frequency of ultrasound produced for a given transducer size, the more directional is the beam.
		A characteristic of the response of both plants and insects to pulsed ultrasound is that the critical exposure parameter appears to be the temporal peak rather than the temporal average of the intensity.
		The spatial distribution of ultrasound fields can be quite complicated depending on such factors as focusing, the radius of the transducer, the wavelength of the ultrasound, the distance from the source, and even on the way in which the element of the transducer is mounted (Zemanek, 1971).
	www.inchem.org /documents/ehc/ehc/ehc22.htm (15125 words)

	kato-pkix-ecc-oef-00.txt
		The identity of a finite field and a specific field element therein may need to be specified, for example, as part of some elliptic curve domain parameters.
		A field element should be represented as a polynomial with integer coefficients, which can be represented as a sequence of the coefficients.
		Informally, the idea is to represent the integer with radix-p positional number system where p is the characteristic of the field, and then convert the each digit to the each coefficient of the polynomial.
	www.potaroo.net /ietf/idref/draft-kato-pkix-ecc-oef (1606 words)

	Finite field - Wikipedia, the free encyclopedia
		Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory.
		The Frobenius automorphism has order n, so that the cyclic group it generates is the full group of automorphisms of the field.
		Finite fields also find applications in coding theory : many codes are constructed as subspaces of vector spaces over finite fields.
	en.wikipedia.org /wiki/Finite_field (749 words)

	Introduction to Semiconductor Superlattices (Site not responding. Last check: 2007-10-16)
		If the field is increased even further, we can have the situation that the lower subband in one quantum well is aligned with the upper one in the adjacent well.
		This field strength is given by the position of the first maximum of the current-voltage characteristic.
		The current-voltage characteristics corresponding to the lines with the letters a, b, c, d and e are presented further below.
	www.lce.hut.fi /~patra/dynamics/sl0 (1632 words)

	Field Guide
		The characteristics in this guide are indicative of a sign's strong influence somewhere in the subject's chart.
		Characteristic behaviors: Honored to serve you food or beverages.
		Characteristic markings: A smile on the face, and a finger pointing at someone else.
	fortuna.home.pipeline.com /cafe-compendium/astro.htm (1498 words)

	APPENDIX J
		The characteristic of a field is its characteristic as a division algebra.
		In this construction, the field elements (or marks in the language of [Dickson 1900]) are residue classes of the integers J modulo the prime p.
		The prime subfield of a finite field is the submodule of the field generated by unity.
	graham.main.nc.us /~bhammel/FCCR/apdxJ.html (6145 words)

	ORDINAL REAL NUMBERS 2 (Site not responding. Last check: 2007-10-16)
		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.
		By the theorem 17 of [Kyritsis C. OR1] the field Q á is a subfield of F á.
	softlab.ntua.gr /~kyritsis/PapersInMaths/InfinityandStochastics/OR2.htm (3131 words)

	[ref] 57 Finite Fields
		, the subfield is the prime field of this characteristic.
		The field is regarded as a vector space (see Vector Spaces) over the given subfield, so this determines the dimension and the canonical basis of the field.
		is the Conway polynomial of the finite field GF(p
	www-gap.dcs.st-and.ac.uk /Manuals/doc/htm/ref/CHAP057.htm (1970 words)

	Matrix Manual: Algeraic Structures
		The characteristic of an integral domain or field is always either 0 or a prime number.
		A field with characteristic 0 is infinite but not all infinite fields have 0 characteristic.
		A field with a finite number of elements, k, is a Galois Field and denoted by GF( k).
	www.ee.ic.ac.uk /hp/staff/dmb/matrix/vector.html (1005 words)

	James Cox Patent (1990): Dipole Accelerating Means And Method
		The means includes a means for generating an alternating electric field extending a first direction, which varies at a selected frequency and which has a predetermined magnitude which is less than the characteristic field ionization potential limit of a particle.
		The alternating magnetic field extends in a second direction at a predetermined angle to and crosses and intercepts the electric field to define a spatial force field region.
		The alternating magnetic field has a frequency which is substantially equal to and is at a predetermined phase angle relative to the alternating electric field and is at a flux density which, when multiplied times the selected frequency, is less than the characteristic field ionization limit of a particle.
	www.padrak.com /agn/JCPATENT.html (374 words)

	Dynamic and Seismic Analysis of Foundations Based on Free Field B-Spline Characteristic Response Histories (Site not responding. Last check: 2007-10-16)
		The characteristic responses are computed in the form of time-dependent flexibility matrices of the medium that are sparse due to the finite duration of the B-Spline excitation signal and the characteristics of the wave propagation.
		Furthermore, the characteristic responses do not depend on the type or wave form of the actual external excitations and the presence of rigid foundations.
		The significance of nonrelaxed boundary conditions and correct representation of the free field is established.
	www.pubs.asce.org /WWWdisplay.cgi?0201645 (289 words)

	PlanetMath: a representation which is not completely reducible
		spans a complementary subspace, but over characteristic 2, these elements are the same.
		Cross-references: complementary subspace, spans, spanned by, generates, identity, characteristic function, homomorphism, subspace, invariant subspace, complementary, constant functions, subrepresentation, obvious, completely reducible, representation, action, functions, regular representation, Maschke's theorem, group, order, divide, characteristic, field, finite group
		This is version 2 of a representation which is not completely reducible, born on 2003-03-24, modified 2005-03-10.
	www.planetmath.org /encyclopedia/ARepresentationWhichIsNotCompletelyReducible.html (167 words)

	ORDINAL REAL NUMBERS 2 (Site not responding. Last check: 2007-10-16)
		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.
		It holds that the rank of the extension F/K is a cofinal order-type with the characteristic of the field F. That is cf(r(F/K))= cf(charF)=cf(char F-char K).
	www.softlab.ntua.gr /~kyritsis/PapersInMaths/InfinityandStochastics/OR2.htm (2766 words)

	Historical Soviet Daily Snow Depth Version 2 (HSDSD)
		If the SDC field is "5", the value was originally derived from humidity data (not snow depth); therefore, SD is artificially set to "0." These data are left in the record to maintain a uniform time series and to indicate the type of original data; however, they can safely be ignored.
		Data fields are fixed length and comma-delimited with a total record length of 34 characters.
		NSIDC added additional flags in the characteristic field to indicate data with errors that were fixed during processing.
	nsidc.org /data/docs/daac/g01092_hsdsd.gd.html (3408 words)

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