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Topic: Algebraic extension

	[No title] (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 variables in M, such that p(x) is true and the set {y in N
	www.informationgenius.com /encyclopedia/a/al/algebraic_extension.html (344 words)

	What Is Mpp Extension (Site not responding. Last check: 2007-10-07)
		Agricultural extension is the application of scientific research and new knowledge to agricultural practices through farmer education.
		The extension is said to be finite or infinite according as the degree is finite or infinite.
		Extension also plays an important part in the philosophy of Spinoza, who claims that substance (that which has extension) can only be limited by substance of the same sort, i.e.
	www.wwwtln.com /finance/205/what-is-mpp-extension.html (1234 words)

	Extension Pole (Site not responding. Last check: 2007-10-07)
		The extension of an object in abstract algebra, such as a group, is the underlying set of the object.
		The extension of a whole ''statement'', as opposed to a word or phrase, is defined (by convention) as its truth-value.
		A filename extension or filename suffix is an extra set of (usually) alphanumeric characters that is appended to the end of a filename to allow computer users (as well as various pieces of software on the computer system) to quickly determine the type of data stored in the file.
	www.wwwtln.com /finance/72/extension-pole.html (2009 words)

	[ref] 61 Algebraic extensions of fields (Site not responding. Last check: 2007-10-07)
		According to Kronecker's construction, the elements of an algebraic extension considered to be polynomials in the primitive element.
		therefore displays elements of an algebraic extension as polynomials in an indeterminate ``a'', which is a root of the defining polynomial of the extension.
		The external representation of algebraic extension elements are the polynomial coefficients in the primitive element
	www.karlin.mff.cuni.cz /asc/network/prirucky/gap/ref/CHAP061.htm (311 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-07)
		Next you consider an algebraic equation in one indeterminate (or unknown) whose coefficients lie in the field F. Another way to describe this equation is a polynomial with coefficients in F set equal to zero.
		This field K is called an algebraic extension of the field F, and is often denoted K = F(a), which is read, "F with a adjoined." It is an algebraic extension because a satisfies and algebraic equation, f(a) = 0.
		The opposite of an algebraic extension is a transcendental extension, which is gotten by adjoining an element b which satisfies no such equation.
	mathforum.org /library/drmath/view/51640.html (465 words)

	Algebraic extension -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-07)
		In (Click link for more info and facts about abstract algebra) abstract algebra, a (Click link for more info and facts about field extension) field extension L/K is called algebraic if every element of L is (Click link for more info and facts about algebraic) algebraic over K, i.e.
		For example, the field extension (Click link for more info and facts about R) R/ (Click link for more info and facts about Q) Q is transcendental, while the field extensions (Click link for more info and facts about C) C/R and Q(√2)/Q are algebraic.
		For instance, the field of all (Root of an algebraic equation with rational coefficients) algebraic numbers is an infinite algebraic extension of the rational numbers.
	www.absoluteastronomy.com /encyclopedia/A/Al/Algebraic_extension.htm (640 words)

	Field (mathematics) - Wikipedia, the free 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].
	xahlee.org /_p/wiki/Field_(mathematics).html (1204 words)

	Algebraic number field - Wikipedia, the free encyclopedia
		In mathematics, an algebraic number field (or simply number field) is a finite (and therefore algebraic) field extension of the rational numbers Q.
		That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q.
		The study of algebraic number fields, and these days also of infinite algebraic extensions of the rational number field, is the central topic of algebraic number theory.
	en.wikipedia.org /wiki/Algebraic_number_field (105 words)

	[No title]
		Moreover, if we fix a finite algebraic extension F of the rationals, the set of all algebraic integers lying in F forms a subring of F called the "ring of integers" of F. Rings of algebraic integers have many properties analogous to those of the usual integers.
		It is a fundamental theorem of algebraic number theory that for any finite algebraic extension of the rationals, the class group is always a finite group.
		So if F is a finite algebraic extension of the rationals, the only extensions K of F for which Gal(K/F) could possibly be equal to the class group of F are those for which Gal(K/F) is abelian (the "abelian extensions of F").
	www-math.mit.edu /~tchow/mathstuff/CFT (1977 words)

	ABSTRACT ALGEBRA ON LINE: Fields
		The field F is said to be an extension field of the field K if K is a subset of F which is a field under the operations of F. Definition.
		Let F be an extension field of K. If the dimension of F as a vector space over K is finite, then F is said to be a finite extension of K. The dimension of F as a vector space over K is called the degree of F over K, and is denoted by [F:K].
		Let F be an extension field of K. The set of all elements of F that are algebraic over K forms a subfield of F. Definition.
	www.math.niu.edu /~beachy/aaol/fields.html (1525 words)

	P-adic number - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-10-07)
		In the algebraic approach, we first define the ring of p-adic integers, and then construct the field of quotients of this ring to get the field of p-adic numbers.
		The real numbers have only a single proper algebraic extension, the complex numbers; in other words, this quadratic extension is already algebraically closed.
		Thus e is a member of the algebraic closure of p-adic numbers for all p.
	encyclopedia.learnthis.info /p/p_/p_adic_number.html (1334 words)

	AlgebraicNumberFields
		Representation of algebraic numbers as elements of a finite extension of rationals.
		Arithmetic within a fixed finite extension of rationals is much faster than arithmetic within the field of all complex algebraic numbers.
		, represented as an element of an extension of rationals of degree 4, is the square of
	documents.wolfram.com /v5/Add-onsLinks/StandardPackages/NumberTheory/AlgebraicNumberFields.html (893 words)

	Algebraic extension (Site not responding. Last check: 2007-10-07)
		For every non-zero polynomial p with coefficients in F, there is an algebraic extension G of F and an x in G such that p(x) = 0.
		The isomorphism is not, in general, unique: the group of automorphisms of F[x] over F is called the Galois group of x.
		Every field is contained in an algebraically closed field (called the algebraic closure), but proving this in general requires some form of the axiom of choice.
	brandt.kurowski.net /projects/lsa/wiki/view.cgi?doc=504 (268 words)

	Galois Theory Glossary
		An extension of fields L/K (this notation does not denote any sort of quotient) is a ring homomorphism K --> L. Such a homomorphism has to be injective, so that K is isomorphic to a subfield of L. It is often convenient to identify K with this subfield.
		A Galois extension is a normal separably algebraic extension of finite degree.
		Let L/K be an extension of fields, and let a in L be algebraic over K. The minimal polynomial of a over K is the unique monic polynomial f(t) in K[t] of least degree such that f(a) = 0.
	www.wra1th.plus.com /Galois/gloss.html (892 words)

	Closure Medical -- Recommendations and Resources (Site not responding. Last check: 2007-10-07)
		The algebraic closure of ''K'' is also the smallest algebraically closed field containing ''K'', because if ''M'' is any algebraically closed field containing ''K'', then the elements of ''M'' which are algebraic over ''K'' form an algebraic closure of ''K''.
		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 transcendental extensions of the rational numbers, e.g.
		In abstract algebra, the concept of integral closure is a generalization of the set of all algebraic integers.
	www.becomingapediatrician.com /health/33/closure-medical.html (1330 words)

	Algebraic and Transcendental (Site not responding. Last check: 2007-10-07)
		Equivalently, an algebraic number is the root of an irreducible polynomial in the rationals.
		Adjoining a finite set of algebraic elements produces a finite extension, and every element in a finite extension is algebraic, hence the extension is algebraic.
		Each extension has dimension p, and if the composite extension were finite, there would not be room for a p dimensional extension for large p.
	www.mathreference.com /fld,algtrans.html (444 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.
		Every projective algebraic set can be written uniquely as a finite irredundant union of irreducible projective algebraic sets.
	www.risberg.ws /Hypertextbooks/Mathematics/Geometry/algebraic.htm (1388 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		Sci., University of St. Andrews, Scotland %Y Copyright (C) 2002 The GAP Group %% \Chapter{Algebraic extensions of fields} If we adjoin a root $\alpha$ of an irreducible polynomial $f \in K[x]$ to the field $K$ we get an algebraic extension $K(\alpha)$, which is again a field.
		{\GAP} therefore displays elements of an algebraic extension as polynomials in an indeterminate ``a'', which is a root of the defining polynomial of the extension.
		The remedy is to multiply the list(s) with the `One' of the extension which will embed all entries in the extension.
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/doc/build/algfld.msk (381 words)

	[No title]
		Any simple algebraic field is isomorphic to the field formed by adjoining an indeterminate, x, to the base field, and then factoring out by the two sided ideal generated by the minimum polynomial of the algebraic element over the base field.
		A chain of degree two extensions is called a square root tower (since the square root of anything in one extension is the addition to make the next higher (simple) extension).
		Since the extension field is merely the base field with the roots appended, the transitivity theorem states that anything in the Galois group moves these roots to each other.
	www.people.virginia.edu /~jba5b/552_midterm.doc (1314 words)

	ABSTRACT ALGEBRA: OnLine Study Guide, Section 6.2 (Site not responding. Last check: 2007-10-07)
		Let F be an extension field of K and let u be an element of F that is algebraic over K. If the minimal polynomial of u over K has degree n, then K(u) is an n-dimensional vector space over K. Definition 6.2.2.
		Let F be an extension field of K. The set of all elements of F that are algebraic over K forms a subfield of F. Definition 6.2.8.
		An extension field F of K is said to be algebraic over K if each element of F is algebraic over K. Proposition 6.2.9.
	www.math.niu.edu /~beachy/abstract_algebra/study_guide/62.html (378 words)

	Introduction to "Taming Wild Extensions" of Lindsay N. Childs
		Galois module theory is the branch of algebraic number theory which studies rings of integers of Galois extensions of number fields as modules over the integral group ring of the Galois group.
		Since wild extensions include all ramified Galois extensions of a local field K containing \Bbb Q_p where the Galois group is a p-group, this was a substantial omission.
		We then give Byott's classification of Galois extensions for which the classical Galois structure is the unique Hopf Galois structure, and survey results on the number of Hopf Galois structures on Galois extensions with Galois group G for various G, including cyclic p-groups {Ko98}, and symmetric, alternating and simple groups {CC99}.
	math.albany.edu:8000 /~lc802/mono.html (1829 words)

	Introduction
		An algebraic set in projected form is given by a polynomial and a tuple of rational functions (specifying the birational mapping).
		An algebraic set in parametric form is given by a tuple of rational functions that parametrizes the algebraic set.
		An algebraic curve is given by places if for each branch passing through a certain point on the algebraic set a tuple of power series that parametrizes the algebraic set around the point is specified.
	www.risc.uni-linz.ac.at /software/casa/Introduction.html (624 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.
		That theory flourished in the latter part of the 19th century, but was approaching its practical limits just as Hilbert, Noether, and others began developing the abstract approach that still dominates modern algebra.
		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)

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