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Topic: Separable extension

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Purely inseparable field extension

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	Separable Extensions (Site not responding. Last check: 2007-10-14)
		The irreducible polynomial p(x) over k is separable if its roots, in the splitting field f/k, are all distinct.
		The extension f/k is separable if every u in f is separable.
		An inseparable extension is algebraic, but not separable.
	www.mathreference.com /fld-sep,intro.html (238 words)

	Field extension - Wikipedia, the free encyclopedia
		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.
		In case the extension is Galois, this automorphism group is called the Galois group of the extension.
		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 (1564 words)

	PlanetMath: separable
		is separable; examples of inseparable extensions include the quotient field
		Cross-references: basis, finitely generated, field extension, variable, rational functions, quotient field, extensions, characteristic, minimal polynomial, algebraic field extension, irreducible, polynomial, splitting field, factors, field, irreducible polynomial
		This is version 9 of separable, born on 2002-01-05, modified 2005-03-05.
	planetmath.org /encyclopedia/SeparablePolynomial.html (99 words)

	Galois Extension Field -- from Wolfram MathWorld
		One is for it to not be a normal extension.
		The field characteristic of such a field must be finite since all polynomials are separable in characteristic zero.
		Moreover, all finite fields are perfect, i.e., all algebraic extensions are separable.
	mathworld.wolfram.com /GaloisExtensionField.html (266 words)

	Separable extension - Wikipedia, the free encyclopedia
		In mathematics, a field extension L/K is separable if it can be generated by adjoining to K a set each of whose elements is a root of a separable polynomial over K.
		Given a finite extension L/K of fields, there is a smallest subfield M of L containing K such that L is a separable extension of M.
		In general an algebraic extension factors as a purely inseparable extension of a separable extension, since the compositum of a family of separable extensions is again separable.
	en.wikipedia.org /wiki/Separable_extension (363 words)

	Basic Proof
		From our usual extension theory, we know the number of possible extensions to k(å1) is just the number of distinct roots in äk of the minimal polynomial g of å1 over k.
		Hence the number of extensions to k(å1) is at most the degree [k(å1):k], and equals that degree iff å1 is separable over k.
		Similarly, the number of further extensions to k(å1,å2) is at most the degree of the field extension [k(å1,å2):k(å1)], and equals that degree iff å2 is separable over k(å1).
	www.physicsforums.com /showthread.php?t=70236 (837 words)

	PlanetMath: Riemann-Hurwitz theorem
		(the different of an extension of Dedekind domains is a fractional ideal).
		be a finite, separable, geometric extension of function fields and suppose the genus of
		Cross-references: formula, genus, geometric extension, separable, generated by, free abelian group, divisor, fractional ideal, Dedekind domains, extension, power, localization, prime ideal, integral closure, quotient field, maximal ideal, discrete valuation ring, prime, separable extension, finite, field, function field
	planetmath.org /encyclopedia/RiemannHurwitzTheorem.html (138 words)

	Springer Online Reference Works
		An extension that is not separable is called inseparable.
		In what follows only algebraic extensions will be considered (for transcendental separable extensions see Transcendental extension).
		The separable extensions form a distinguished class of extensions, that is, in a tower of fields
	eom.springer.de /s/s084470.htm (187 words)

	[No title]
		Abstract.This paper establishes analogues of the the classical decomposi- tions of a normal field extension in the context of graded integral doma* *ins with an unstable action of the mod p Steenrod algebra.
		Interpretation of the separability condition is harder, but algebraic exampl* es at all primes and topological examples at the prime 2 show that it is a necessa *ry complication.
		L is a separable extension of graded fields, then L has a u* *nique Ap action respecting the Cartan formula and such that i is a map of Ap-algebras.
	www.math.purdue.edu /research/atopology/Wilkerson/newring.txt (6367 words)

	S.O.S. Mathematics CyberBoard :: View topic - Question about perfect fields
		Separable element: An element a in an extension K of F is called separable over F if it satisfies a polynomial over F having no multiple roots.
		Separable extension: An extension K of F is called separable over F if all its elements are separable over F. Perfect field: A field F is called perfect if all finite extensions of F are separable.
		Thus, some finite extension K of F is not separable, or some element in K satisfies a polynomial over F having multiple roots.
	www.sosmath.com /CBB/viewtopic.php?p=121135&sid=685d68efdda99006d14eddba4c23e083 (403 words)

	Separable Extension -- from Wolfram MathWorld
		is a separable extension since the minimal polynomial of
		In fact, in field characteristic zero, every extension is separable, as is any finite extension of a
		It is a bit more complicated to describe a field which is not separable.
	mathworld.wolfram.com /SeparableExtension.html (123 words)

	The Fundamental Theorem of Algebra (Site not responding. Last check: 2007-10-14)
		To show C is closed it is sufficient to show there are no finite dimensional extensions of C. If e is such an extension it is also a finite separable extension over R. Separable, because the characteristic of R is 0.
		Let l be the extension of R that is fixed by h.
		This takes place inside a galois extension, so the dimension of l/R is the index of h in g, which is odd.
	www.mathreference.com /fld-sep,fta.html (515 words)

	A SPATIAL EXTENSION OF CIELAB FOR DIGITAL COLOR IMAGE REPRODUCTION 1
		We describe a spatial extension to the CIELAB color metric that is useful for measuring color reproduction errors of digital images.
		The calculation is pattern-color separable because the color transformation does not depend on the image's spatial pattern, and the spatial convolution does not depend on the image's color.
		The separability of the pattern and color stages makes it straightforward to apply the spatial extension to other color difference calculations.
	white.stanford.edu /~brian/scielab/scielab3/scielab3.html (1239 words)

	[No title]
		MU is an S[BU]-Hopf-Galois extension of commutative S-algebras, with coaction fi :MU !
		As in the discrete algebraic case, the characteristic property is that C is separable over A, and in Section* * 9.1 we develop the basic theory of separable extensions of S-algebras.
		The idea of Hopf-Galois extensions is to replace the action by the Galois gr* *oup G on a commutative A-algebra B by a coaction by the functional dual DG+ = F (G+, S) of the Galois group, which is a commutative Hopf S-algebra.
	www.math.purdue.edu /research/atopology/Rognes/galois.txt (18535 words)

	Mathenomicon.net : Fields and extensions (Site not responding. Last check: 2007-10-14)
		Each line represents an extension, and the number labelling the line is the degree.
		Since any finite extension can be built up using a sequence of simple extensions, the following result can be used to compute the separable degree of any finite extension.
		The natural progression from this point is the study of Galois Theory, which considers the properties of automorphism groups of normal extensions.
	www.cenius.net /refer/articles/f/fieldsandextension_ency/fieldsandextension_ency.html (671 words)

	[No title]
		] An element a is said to be separable over a field F if it is algebraic over F and if the extension field of F generated by a is a separable extension of F.
		] The separation of liquids or gases in a mixture, as by distillation or extraction.
		A machine for separating materials of different specific gravity by means of water or air.
	www.accessscience.com /Dictionary/S/S17/DictS17.html (2741 words)

	Separable Extension
		首先我們會發現幾乎在大學代數中談的 extension 都是 separable extension, 然後我們會進一步討論 separable extension 重要的性質.
		Lemma 3.4.2 假設 L/K 是一個 finite separable extension 且 F 是 L/K 的 intermediate field.
		Theorem 3.4.5 假設 L/K 是一個 finite extension 且 N 是一個 L 的 extension 滿足 N/K 是 finite normal extension.
	math.ntnu.edu.tw /~li/galois-html/node15.html (785 words)

	Electrical Connectors - Swivel - National Hardware Show - extension cord connectors (Site not responding. Last check: 2007-10-14)
		The fast separable feature provides a practical means for a construction worker or assembly plant worker to switch between power tools.
		For 2 wire power tools, the 3 contact swiveling, separable electrical connectors can still be used by simply hardwiring only 2 wires to 2 of the contacts.
		Also, these separable rotating electrical connectors can be spliced into popular extension cords in various types of construction sites and/or assembly plants around the world.
	www.power-delivery.com /connectors.html (464 words)

	ABSTRACT ALGEBRA ON LINE: Galois Theory
		An algebraic extension field F of K is called separable over K if the minimal polynomial of each element of F is separable.
		Let F be a finite extension of the field K. If F is separable over K, then it is a simple extension of K. The fundamental theorem of Galois theory
		Let K be a field of characteristic zero, and let E be a radical extension of K. Then there exists an extension F of E that is a normal radical extension of K. Theorem.
	www.math.niu.edu /~beachy/aaol/galois.html (1898 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		Here the most intriguing problem is whether K (the space of compact operators on a separable Hilbert space) is complemented in every separable operator space containing it (this is a "non-commutative" version of a classical result of Sobczyk).
		A related issue is giving a complete description of separable locally reflexive operator spaces which are completely complemented in every separable locally reflexive operator superspace (an operator space counterpart of Zippin's characterization of the space of convergent sequences).
		The PI plans to determine the cardinality of the set of n-dimensional subspaces of maximal spaces, as well as the structure of n-dimensional subspaces of the dual of a 2n-dimensional commutative C*-algebra (in the spirit of Kashin's work).
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9970369.txt (322 words)

	Separable Subfield (Site not responding. Last check: 2007-10-14)
		If u and v are separable then f is the splitting field for a set of separable polynomials, and is galois.
		A galois extension is separable, so everything in f is separable.
		The set of elements e in f that are separable over k are closed under addition and multiplication, and form an intermediate field extension.
	www.mathreference.com /fld-sep,subf.html (73 words)

	ADFS::HD4.$.Work.courses.98-99.Galois.Notes.N3
		We call a polynomial separable if the highest common factor of the polynomial and its formal derivative has degree zero.
		It follows from the arguments above that a polynomial is separable if it has no repeated roots in any extension of the field in which its coefficients lie.
		The assumption of separability tells us that the numerator and denominators of these fractions are nonzero, and because there are only finitely many such expressions the assumption that
	www.wra1th.plus.com /Galois/N3.html (265 words)

	ABSTRACT ALGEBRA: OnLine Study Guide, Section 8.3
		Let F be an algebraic extension of the field K. Then F is said to be a normal extension of K if every irreducible polynomial in K[x] that contains a root in F is a product of linear factors in F[x].
		Let F be the splitting field of a separable polynomial over the field K, and let E be a subfield such that K
		Let F be a finite, normal, separable extension of the field K. Suppose that the Galois group Gal (F/K) is isomorphic to D
	www.math.niu.edu /~beachy/abstract_algebra/study_guide/83.html (629 words)

	Springer Online Reference Works
		A field extension that is not algebraic (cf.
		is called separably generated, and the transcendence basis of
		Such an extension is uniquely determined for any derivation if and only if the extension
	eom.springer.de /T/t093620.htm (173 words)

	separable - OneLook Dictionary Search
		-separable, -separable, separable, separable, separable, separable : PlanetMath Encyclopedia [home, info]
		Phrases that include separable: separable affix, separable extension, linearly separable, separable closure, separable space, more...
		Words similar to separable: dissociable, separability, separableness, severable, more...
	www.onelook.com /?w=separable (196 words)

	ADFS::HD4.$.Work.courses.98-99.Galois.Notes.N4
		A Galois extension is a finite normal separable extension.
		It follows from this that intermediate extensions are conjugate if and only if the corresponding subgroups of the Galois group are conjugate.
		is not a Galois extension unless it is normal, and this happens precisely when it equals all its conjugates.
	www.wra1th.plus.com /Galois/N4.html (278 words)

	Algebraic Number Theory (Site not responding. Last check: 2007-10-14)
		We are therefore interested in integral extensions of Z (the integers), and field extensions of Q (the rationals).
		A global field is a finite separable extension of Q or of Z
		A number field is a finite extension of Q. Since Q has characteristic 0, extensions of q are automatically separable.
	www.mathreference.com /an,intro.html (185 words)

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