
	Where results make sense

Topic: Algebraic closure

Related Topics

Algebraically closed field

Algebraic number

	Closure (mathematics) - Wikipedia, the free encyclopedia
		The closure is idempotent: the closure of the closure equals the closure.
		In linear algebra, the linear span of a set X of vectors is the closure of that set; it is the smallest subset of the vector space that includes X and is a subspace.
		In algebra, the closure of a set S under a binary operation is the smallest set C(S) that includes S and is closed under the binary operation.
	en.wikipedia.org /wiki/Closure_(mathematics) (440 words)

	Algebraic number - Wikipedia, the free encyclopedia
		The sum, difference, product and quotient of two algebraic numbers is again algebraic, and the algebraic numbers therefore form a field, called the algebraic closure of the field of algebraic numbers.
		In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.
		The name algebraic integer comes from the fact that the only rational numbers which are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers.
	en.wikipedia.org /wiki/Algebraic_number (609 words)

	PlanetMath: algebraic extension (Site not responding. Last check: 2007-10-07)
		In general, a finite extension of fields is an algebraic extension.
		Cross-references: finite, a finite extension of fields is an algebraic extension, algebraic closure, pi, transcendental number, field extension, root, algebraic, fields, extension
		This is version 3 of algebraic extension, born on 2003-09-11, modified 2003-09-28.
	planetmath.org /encyclopedia/AlgebraicExtension.html (95 words)

	Integral closure - Wikipedia, the free encyclopedia
		In abstract algebra, the concept of integral closure is a generalization of the set of all algebraic integers.
		The integral closure of Z in the complex numbers C is the set of all algebraic integers.
		See also algebraic closure; this is a special case of integral closure when R and S are fields.
	www.wikipedia.org /wiki/Integral_closure (485 words)

	Field (mathematics) - Open Encyclopedia (Site not responding. Last check: 2007-10-07)
		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 associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers.
		Fields are important objects of study in algebra since they provide the proper generalization of number domains, such as the sets of rational numbers, real numbers, or complex numbers.
		An algebraic extension of a field F is the smallest field containing F and a root of an irreducible polynomial p(x) in F[x].
	open-encyclopedia.com /Field_(mathematics) (1226 words)

	Algebraically closed field (Site not responding. Last check: 2007-10-07)
		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.
	hallencyclopedia.com /Algebraically_closed_field (509 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.
		The integral closure of A in B is the subset of B consisting of elements integral over A; the subring A of B is said to be integrally closed in B if it is its own integral closure in B. Theorem (4.23).
		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.
	www.math.harvard.edu /~elkies/M250.04/closure.html (614 words)

	Algebraic number (Site not responding. Last check: 2007-10-07)
		If an algebraic number satisifies such an equation as given above with a polynomial of degree n and not such an equation with a lower degree, then the number is said to be analgebraic number of degree n.
		In fact, it is the smallestalgebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.
		Both the notions of algebraic number and algebraic integer may be usefully generalized to fields other than the complexnumbers; see algebraic extension and integral closure.
	www.therfcc.org /algebraic-number-19429.html (364 words)

	Symmetric function - Wikipedia, the free encyclopedia
		are in an algebraic closure, we have the sum of all the α
		A great deal of attention was paid, in older algebra textbooks, to algorithmic procedures expressing the procedural content of this (which has been stated as an existence theorem but has computational content).
		in a formal power series ring; here passage to the algebraic closure is the theory of Puiseux expansions in fractional powers, and the Newton polygon is a device for computing the required exponents.
	en.wikipedia.org /wiki/Symmetric_function (537 words)

	Algebraic extension (Site not responding. Last check: 2007-10-07)
		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.
		Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p(x) is true and the set
	www.encyclopedia-1.com /a/al/algebraic_extension.html (353 words)

	Algebraic closure -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-07)
		It is one of many (Termination of operations) closures in mathematics.
		The algebraic closure of a field K has the same (Click link for more info and facts about cardinality) cardinality as K if K is infinite, and is (Click link for more info and facts about countably infinite) countably infinite if K is finite.
		The algebraic closure of the field of (An integer or a fraction) rational numbers is the field of (Root of an algebraic equation with rational coefficients) algebraic numbers.
	www.absoluteastronomy.com /encyclopedia/a/al/algebraic_closure.htm (263 words)

	Closure Medical -- Recommendations and Resources (Site not responding. Last check: 2007-10-07)
		The definition of a point of closure is closely related to the definition of a limit point.
		A point of closure which is not a limit point is an isolated point.
		I phrased the bit about density as a ''fact relating'' it to closure (expressed with "iff") rather than as a ''definition in terms of'' closure (expressed with "if"), since there are alternative definitions of dense sets.
	www.becomingapediatrician.com /health/33/closure-medical.html (1330 words)

	Algebraic Closure articles and news from Start Learning Now (Site not responding. Last check: 2007-10-07)
		In mathematics, particularly abstract algebra, an algebraic closure of a field (mathematics)field K is an algebraic extension of K that is algebraically closed fieldalgebraically closed.
		Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixed point (mathematics)fixes every member of K.
		The algebraic closure of a field K has the same cardinal numbercardinality as K if K is infinite, and is countably infinite if K is finite.
	www.startlearningnow.com /algebraic%20closure.htm (419 words)

	P-adic number - Wikipedia, the free encyclopedia
		The real numbers have only a single proper algebraic extension, the complex numbers; in other words, this quadratic extension is already algebraically closed.
		By contrast, the algebraic closure of the p-adic numbers has infinite degree.
		Thus e is a member of the algebraic closure of p-adic numbers for all p.
	www.wikipedia.com /wiki/p-adic+numbers (2053 words)

	Algebraic closure: Definition and Links by Encyclopedian.com - All about Algebraic closure (Site not responding. Last check: 2007-10-07)
		In mathematics, an algebraic closure of a field K is an algebraic extension of K which is algebraically closed.
		Using Zorn's Lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure is unique in the following sense: if L and M are algebraic closures of a field K, then there is an isomorphism f : L
		There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers; these are the algebraic closures of transcendantal extensions of the rational numbers.
	www.encyclopedian.com /al/Algebraic-closure.html (354 words)

	Algebraic closure (Site not responding. Last check: 2007-10-07)
		The fundamental theorem of algebra states that the algebraic closure of the field of real numbers is the fieldof complex numbers.
		The algebraic closure of the field of rational numbers is thefield of algebraic numbers.
		There are many countable algebraically closed fields within the complex numbers, and strictly containing the field ofalgebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers, e.g.
	www.therfcc.org /algebraic-closure-73935.html (318 words)

	RELATIONAL CLOSURE:
		This concept of algebraic closure of a transformation system illustrates some of the important features of the concept we are looking for, but it is not general enough for the task of modelling complex systems by picking out all the relevant distinctions.
		This means that the complement of a closure (in the sense of complete absence of the missing elements, not in the sense of incomplete presence) can in general also be interpreted as a closure.
		Fourth, certain types of closure may be seen as generalizations or specializations of other types of closure, in the sense that a more general closure is characterized by less strict requirements, and hence is less distinction-reducing or redundancy-generating.
	pespmc1.vub.ac.be /papers/RelClosure.html (3936 words)

	Galois Theory Glossary (Site not responding. Last check: 2007-10-07)
		An element a of a field L is algebraic of degree n over a subfield K if there is an irreducible polynomial f(t) in K[t] of degree n such that f(a) = 0.
		A field K is algebraically closed if every polynomial in K[t] of degree greater than zero has a root in K. Equivalently every polynomial in K[t] of degree greater than zero splits in K[t] into a product of linear factors.
		An element a in a field L is a separably algebraic element over a subfield K if there is a polynomial f(t) in K[t] such that f(a) = 0 and f'(a) is nonzero.
	www.wra1th.plus.com /Galois/gloss.html (892 words)

	JISE Vol. 16 No. 4 #2 (Site not responding. Last check: 2007-10-07)
		Operands as well as the results of operations in the proposed algebra are formally characterized as pairs of sets - a set of objects capturing the states and a set of message expressions comprised of sequences of messages modeling the object behavior.
		The closure property is achieved in a natural way by letting the results of operations possess the same characteristics as do the operands in an algebra expression.
		Furthermore, the result of an object algebra expression is shown to have the characteristics of a class whose superclass/subclass relationships with its operand class(es) can be established, thus providing a mechanism to properly and persistently place it in the class lattice (schema).
	www.iis.sinica.edu.tw /JISE/2000/200007_02.html (246 words)

	Closed Field
		The closure of k is a minimal field extension of k that splits every polynomial.
		Let f be the closure of k, and suppose f is not closed.
		To find the closure of the rationals, adjoin all the roots of all the polynomials with rational coefficients, giving a subfield of the complex numbers that is closed.
	www.mathreference.com /fld,closed.html (687 words)

	On the topological cyclic homology of the algebraic closure of a local field (Site not responding. Last check: 2007-10-07)
		On the topological cyclic homology of the algebraic closure of a local field
		Let V be a complete discrete valuation ring with quotient field K of characteristic 0 and perfect residue field k of odd characteristic p.
		This was used to evaluate the p-adic K-groups of the field K. In the present paper, we completely determine the structure of the homotopy groups with Z/p^v-coefficients, for all v, of the colimit
	www-math.mit.edu /~larsh/papers/020 (180 words)

	DYNAMIC EVALUATION AND COMPUTING WITH PARAMETERS (Site not responding. Last check: 2007-10-07)
		Dynamic evaluation was first applied to computations with algebraic numbers, implemented as the dynamic algebraic closure of a field, first written in Reduce (known as the D5 system), and later in Axiom [DDD, DD, Du].
		The dynamic constructible closure of a field allows only the equality test, but here one deals with parameters in a more general sense than the algebraic one.
		As in the algebraic case, the gcd is the tool used in answering equality tests.
	www.aldor.org /docs/reports/i95dynev/node2.html (309 words)

	ADFS::HD4.$.Work.courses.98-99.Galois.Notes.N2 (Site not responding. Last check: 2007-10-07)
		Theorem: A composite of algebraic extensions is algebraic.
		The subring of algebraic numbers (those complex numbers which are roots of polynomials with rational coefficients) is the algebraic closure of
		The point about algebraic closure is that every field can be considered a subfield of an algebraically closed field, and polynomials can always be split up into linear factors over that.
	www.wra1th.plus.com /Galois/N2.html (437 words)

	The Ultimate Algebraic function - American History Information Guide and Reference
		That is, F is an implicit function that solves an algebraic equation.
		The class of algebraic functions contains all rational functions, but is larger.
		In fact in terms of abstract algebra it is the algebraic closure of the field of rational functions, for any fixed set of indeterminates.
	www.historymania.com /american_history/Algebraic_function (118 words)

Try your search on: Qwika (all wikis)