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Topic: Algebraically closed

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	Algebraically closed field - Wikipedia, the free encyclopedia
		By contrast, the field of complex numbers is algebraically closed: this is stated by 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.
		Also, the field of algebraic numbers is the algebraic closure of the field of rational numbers.
	en.wikipedia.org /wiki/Algebraically_closed (237 words)

	Algebraically closed and existentially closed substructures in categorical context (Site not responding. Last check: 2007-11-06)
		Algebraically closed and existentially closed substructures in categorical context
		We investigate categorical versions of algebraically closed (= pure) embeddings, existentially closed embeddings, and the like, in the context of locally presentable categories.
		Related preservation theorems are obtained, and a new proof of the main result of Rosicky, Adamek and Borceux, characterizing $\lambda$-injectivity classes in locally $\lambda$-presentable categories, is given.
	www.tac.mta.ca /tac/volumes/12/9/12-09abs.html (84 words)

	New Simple Lie algebras: Melikyan Algebras (Site not responding. Last check: 2007-11-06)
		It should be mentioned that the modular Lie algebras of classical type are exactly the Lie algebras of simple algebraic groups and, thus, closely related to the theory of simple finite groups.
		The rigidity of the Lie algebras L(m,n) under filtered deformations is established by M. Kuznetsov in [16].
		Kostrikin and A. umadildaev [14] is the construction of Melikyan algebras as deformations of certain Hamiltonian Lie algebras.
	www.nccu.edu /artsci/math/melikyan/res/node2.html (1082 words)

	PlanetMath: integrally closed
		is said to be integrally closed (or normal) if it is integrally closed in its fraction field.
		See Also: integral closure, algebraic closure, algebraically closed
		This is version 11 of integrally closed, born on 2002-04-23, modified 2004-06-18.
	planetmath.org /encyclopedia/IntegrallyClosed.html (94 words)

	Algebraically closed field (Site not responding. Last check: 2007-11-06)
		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.
		By contrast, the field of complex numbers is algebraically closed, which is the content of the fundamental theorem of algebra.
		Every field which is not algebraically closed can be formally extended by adjoining roots of polynomials without zeros.
	brandt.kurowski.net /projects/lsa/wiki/view.cgi?doc=21 (138 words)

	Algebraic Geometry
		Algebraic geometry is the study of geometric objects by means of algebra.
		Two (structured) algebraic sets are isomorphic if and only if their coordinate rings are isomorphic.
		Under the bijection of the proposition, nonempty irreducible closed algebraic sets correspond to nonirrelevant proper homogeneous prime ideals.
	www.risberg.ws /Hypertextbooks/Mathematics/Geometry/algebraic.htm (1388 words)

	Algebraically closed field (Site not responding. Last check: 2007-11-06)
		function minus its orthogonal projection onto the closed linear span of the score functions for the...
		In mathematics, 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.
		As an example, the field of real numbers is not algebraically closed, because the polynomial
	hallencyclopedia.com /Algebraically_closed_field (509 words)

	Creation of Elements
		The usual way of creating elements within an algebraically closed field A is by coercion from the base field into A, or by construction of roots of polynomials over A (and this may be done indirectly via other functions).
		Given a polynomial f over an algebraically closed field A, or given a polynomial f over some subring of A together with A itself, this function computes all roots of f in A, and returns a sorted sequence of tuples (pairs), each consisting of a root of f in A and its multiplicity.
		A square root always exists since the field is algebraically closed, and the return value is invariable.
	wwwmaths.anu.edu.au /research.groups/aat/htmlhelp/text770.htm (1126 words)

	Algebraically Closed Fields
		Algebraically closed fields (ACF's) have the property that they always contain all the roots of any polynomial defined over them.
		It is not possible to construct explicitly the closure of a field, but the system works by automatically constructing larger and larger algebraic extensions of an original base field as needed during a computation, thus giving the illusion of computing in the algebraic closure of the base field.
		The new system avoids factorization over algebraic number fields when possible, and automatically splits the defining polynomials of a field when factors are found.
	magma.maths.usyd.edu.au /magma/Features/node131.html (305 words)

	Fundamental theorem of algebra - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-11-06)
		The fundamental theorem of algebra (now considered something of a misnomer by many mathematicians) states that every complex polynomial of degree n has exactly n zeroes, counted with multiplicity.
		This shows that the field of complex numbers, unlike the field of real numbers, is algebraically closed.
		Find a closed disk D of radius r centered at the origin such that
	encyclopedia.learnthis.info /f/fu/fundamental_theorem_of_algebra_1.html (471 words)

	Learn more about Hypercomplex number in the online encyclopedia. (Site not responding. Last check: 2007-11-06)
		Whereas complex numbers can be viewed as points in a plane, hypercomplex numbers can be viewed as points in some higher-dimensional Euclidean space (4 dimensions for the quaternions, 8 for the octonions, 16 for the sedenions).
		But none of these extensions forms a field, essentially because the field of complex numbers is algebraically closed - see fundamental theorem of algebra.
		The Clifford algebras are another family of hypercomplex numbers.
	www.onlineencyclopedia.org /h/hy/hypercomplex_number.html (203 words)

	Algebraic and integral closures
		The algebraic closure of F in K is the subset of K consisting of elements algebraic over F. The subfield F of K is said to be algebraically closed in K if it is its own algebraic closure in K. Theorem.
		Let A be a subring of B. The integral closure of A in B is a ring, and is integrally closed in B. The proof is much the same as for algebraic closures.
		It turns out that if A is integrally closed in its own fraction field F, then any A-integral element u of an extension field of F has a minimal polynomial in A[X], and thus satisfies a monic equation over A of degree [F(u):F].
	www.math.harvard.edu /~elkies/M250.04/closure.html (614 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Define a thick subcategory to be a full subcategory of the category of finite-dimensional B-modules that is closed under summands and, if two out of three modules in a short exact sequence are in it, so is the third.
		A similar classification has been obtained by the current authors when B is a finite subalgebra of the mod 2 Steenrod algebra, with scalars extended to the algebraic closure of Z/2.
		In the present paper, we eliminate the annoying requirement that the field be algebraically closed.
	hopf.math.purdue.edu /Hovey-Palmieri/galois.abstract (220 words)

	[No title]
		Recently, in the case B = kG where k is an algebraically closed field and G is a finite group, Benson, Carlson, and Rickard [BCR96 ] have extended the definition of cohomological variety to infin* *itely generated modules, and they have proved that these new varieties satisfy many of the same properties.
		In the setting of sub-Hopf algebras of A, quasi-elementary Hopf algebras are accessible: a sub-Hopf algebra Q of A is quasi-elementary if and only if it is isomorphic, as an ungraded algebra, to the mod 2 group algebra of an elementary abelian 2-group.
		The group algebra of an elementary abelian p-group is hereditary, as are the quasi-elementary sub* -Hopf algebras* of the mod 2 Steenrod algebra.
	hopf.math.purdue.edu /Hovey-Palmieri/quillen.txt (13317 words)

	New Simple Lie algebras: Melikian Algebras
		Later on several generalizations of the Witt algebra were constructed (by Jacobson, Zassenhaus, Frank, Kaplansky).
		The well-known conjecture of Kostrikin-Shafarevich [KS1], that any non-classical simple Lie algebra over an algebraically closed field F of characteristic p>5 is Lie algebra of Cartan type, has been proved by R.
		Later on A. Premet [P1] proved that any non-classical simple Lie p-algebra over an algebraically closed field of characteristic p=5 with a torus of maximal dimension 2 is either of Cartan type or is isomorphic to the Melikian algebra L(1,1), which I constructed in [M1], [M2].
	www.cs.uwm.edu /~melikian/res/node2.html (341 words)

	Varieties, Ideals, Nullstellensatz
		This leads to the famous Nullstelensatz, a basic theorem in commutative algebra, on which much of algebraic geometry over algebraically closed fields is based.
		Some other ``strange'' things happen over fields, which are not algebraically closed.
		(or any algebraically closed fields) it is a pair of intersecting lines.
	mathcircle.berkeley.edu /BMC3/alg-geom/node1.html (465 words)

	Spectrum
		In linear algebra, the eigenvectors (from the German eigen meaning "inherent, characteristic") of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves.
		In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R.
		The maximal ideals of R correspond to the points of A (because K is algebraically closed), and the prime ideals or R correspond to the subvarieties of A (an algebraic set is called irreducible or a variety if it cannot be written as the union of two proper algebraic subsets).
	www.websters-online-dictionary.com /definition/english/Sp/Spectrum.html (7719 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Show that any two curves over an algebraically closed field are homeomorphic.
		Show that both forms of the Nullstellensatz fail over fields that are not algebraically closed.
		Prove that the algebraic sets are the closed subsets of a topology on
	odin.mdacc.tmc.edu /~krc/agathos/exer.html (567 words)

	Introduction (Site not responding. Last check: 2007-11-06)
		In the latter case, for power series rings the fixed precision is absolute, while for Laurent and Puiseux series ring the fixed precision is relative.
		The free precision rings most closely resemble the mathematical objects R[[x]] and R((x)); elements in these free rings and fields carry their own precision with them.
		Operations usually return results to a precision that is maximal given the input (and the nature of the operation).
	magma.maths.usyd.edu.au /magma/htmlhelp/text766.htm (1006 words)

	[No title]
		We note that irreducibility is really a property of closed sets in the Zariski topology, and does not really involve any other aspects of varieties.
		which is not the union of finitely many irreducible closed subsets, and let S be the collection of closed subsets which are not finite unions of irreducibles.
		This implies that S is a union of finitely many irreducible closed subsets, thus contradicting the hypothesis that S is nonempty and thereby proving the first part of the theorem.
	www.math.umn.edu /~roberts/math8203/irred_var.html (839 words)

	NeuroCOLT: Neural Networks and Computational Learning Theory (Site not responding. Last check: 2007-11-06)
		Blum, Cucker, Shub and Smale have shown that the problem ``P = NP ?'' has the same answer in all algebraically closed fields of characteristic 0.
		We generalize this result to the polynomial hierarchy: if it collapses over an algebraically closed field of characteristic 0, then it must collapse at the same level over all algebraically closed fields of characteristic 0.
		The main ingredient of their proof was a theorem on the elimination of parameters, which we also extend to the polynomial hierarchy.
	www.neurocolt.com /abs/1998/abs98007.html (118 words)

	Definability of Geometric Properties in Algebraically Closed Fields - Chapuis, Koiran (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		We conjecture that the same result holds for the closed subsets of C 2.
		Keywords: definability, constraint databases, model theory, algebraically closed fields.
		72 the combinatorial and algebraic complexity of quantifier-eli..
	citeseer.ist.psu.edu /chapuis98definability.html (554 words)

	A New Program for Computing the P-Linear System Cardinality that Determines the Group of Weil Divisors of a Zariski ...
		Previously an algorithm and associated computer program for determining the group of Weil divisors of a normal Zariski surface, Xg, given by zp=g(x,y), where p>0 is the characteristic of a fixed algebraically closed field, k, containing g(x,y), has been presented.
		The divisor class group, which is an abelian group and a geometric invariant, assists in the classification of algebraic surfaces over a fixed algebraically closed field.
		Problems with p values up to 17 were calculated and compared with known values for divisor class groups of these types as well as the examples in Blass.
	library.wolfram.com /infocenter/Articles/1039 (251 words)

	Definable sets in algebraically closed valued fields. Part I: elimination of imaginaries (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Abstract: It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain cosets in K of certain definable R-submodules of K (for all n 1).
		The proof involves the development of a theory of independence for unary types, which play the role of 1-types, followed by an analysis of germs of definable functions from unary sets to the sorts.
		1 Canonical forms of de nable subsets of algebraically closed..
	citeseer.ist.psu.edu /haskell02definable.html (316 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		ALGEBRA SEMINAR Speaker: Professor Makar-Limanov Title :What are the algebraically closed skewfields and what are they good for?
		Time : 12:30, Thursday,June 27 Place : Amado Building, room 619 Abstract Everybody knows that the field of complex numbers is algebraically closed.
		Even more, most of us know that any field can be embedded in an algebraically closed one.
	www.math.technion.ac.il /~techm/20020627123020020627mak (87 words)

	[No title]
		The restrictive school of thought says that you can avoid the difficulties posed by examples (v) and (vi) by considering only algebraically closed fields.
		Nevertheless, we will acquiesce (at least temporarily), and work over algebraically closed fields for the remainder of this chapter.
		Remark The algebraic sets really do form the closed sets of a topology, but it does not fully reflect the structure of affine space.
	odin.mdacc.tmc.edu /~krc/agathos/curves.html (438 words)

	Introduction
		Having a dense orbit in V is a necessary condition for a linear algebraic group over an algebraically closed field (such as
		In a lecture at the AMS/LMS meeting in July 1992, Martin Liebeck mentioned the above G-module V as one of the open cases in a prospective classification of irreducible modules for almost simple algebraic groups over an algebraically closed field of positive characteristic for which there are a finite number of orbits on points.
		The search may be put in the wider context of finding all pairs of subgroups in almost simple algebraic groups over algebraically closed fields for which there are only finitely many double cosets.
	www.cecm.sfu.ca /organics/papers/acohen/paper/html/node1.html (761 words)

	38a (Site not responding. Last check: 2007-11-06)
		Remark: A simple criterion for an idempotent semifield to be algebraically closed is proved in the paper of G. Shpiz ``Solving algebraic equations in idempotent semifields'', Uspekhi Mat.
		For example, some standard linear function spaces and all the Banach lattices generate algebraically closed idempotent semifields; see, e.g., the paper of G.L. Litvinov, V.P. Maslov, and G.B. Shpiz ``Idempotent functional analysis: an algebraic approach'', Math.
		It would be useful to define tropical/idempotent versions of such notions as algebraic equations and ideals of affine algebraic varieties in such a way that points and subvarieties correspond to analogs of ideals.
	www.aimath.org /WWN/amoebas/articles/html/38a (299 words)

	MUG: Irreducibility in algebraically closed fields with char > 0 (14.3.00)
		Irreducibility in algebraically closed fields with char > 0 (14.3.00)
		I want to test if a polynomial in two variables is or not irreducible over the algebraic closure of F_2, the Galois Field with two elements.
		The trick is to note that any polynomial that factors over the algebraic closure of GF(2) must also factor over GF(2^n) for some n.
	www.math.rwth-aachen.de /mapleAnswers/html/961.html (462 words)

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