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# Topic: Algebraically closed field

###### In the News (Thu 20 Jun 19)

 PlanetMath: separably algebraically closed field has characteristic 0, it is separably algebraically closed if and only if it is algebraically closed. Cross-references: extension, purely inseparable, positive, algebraically closed, characteristic, algebraic closure, separable, field This is version 3 of separably algebraically closed field, born on 2006-06-10, modified 2007-07-03. planetmath.org /encyclopedia/SeparablyAlgebraicallyClosed.html   (99 words)

 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_field   (237 words)

 Field - Wikipedia A field, in abstract algebra, is an algebraic system of elements in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed without leaving the system and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. Fields are important objects of study in abstract algebra since they provide the proper generalization of number domains, such as the sets of rational numbers or real numbers. For a given field F, the set F(X) of rational functions in the variable X with coefficients in F is a field; this is defined as the set of quotients of polynomials with coefficients in F. nostalgia.wikipedia.org /wiki/Field   (1434 words)

 PlanetMath: algebraically closed Moreover, any two algebraic closures a field are isomorphic as fields, but not necessarily canonically isomorphic. Cross-references: isomorphic, axiom of choice, algebraic, extension field, root, polynomial, field This is version 5 of algebraically closed, born on 2002-01-21, modified 2007-06-02. planetmath.org /encyclopedia/AlgebraicallyClosed.html   (78 words)

 Algebraically closed field -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-12) By contrast, the field of (A number of the form a+bi where a and b are real numbers and i is the square root of -1) complex numbers is algebraically closed: this is stated by the (Click link for more info and facts about fundamental theorem of algebra) fundamental theorem of algebra. Each field's algebraic closure is unique (Click link for more info and facts about up to) up to a non-canonical ((biology) similarity or identity of form or shape or structure) isomorphism. Also, the field of (Root of an algebraic equation with rational coefficients) algebraic numbers is the algebraic closure of the field of (An integer or a fraction) rational numbers. absoluteastronomy.com /encyclopedia/a/al/algebraically_closed_field.htm   (249 words)

 Complex number - LearnThis.Info Enclyclopedia   (Site not responding. Last check: 2007-10-12) In mathematics, the term "complex" when used as an adjective means that the field of complex numbers is the underlying number field considered. This is known as the Fundamental Theorem of Algebra, and shows that the complex numbers are an algebraically closed field. Indeed, the complex number field is the algebraic closure of the real number field. encyclopedia.learnthis.info /c/co/complex_number.html   (2277 words)

 Algebraically closed field: Definition and Links by Encyclopedian.com - All about Algebraically closed field   (Site not responding. Last check: 2007-10-12) 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. www.encyclopedian.com /al/Algebraically-closed.html   (190 words)

 Scientific Commons: Patrick Le Meur Let A be a basic connected finite dimensional algebra over an algebraically closed field, let G be a group, let T be a basic tilting A-module and let B the endomorphism algebra of T. Under a... Let A be a basic and connected finite dimensional algebra over an algebraically closed field, let G be a group, let T be a basic tilting A-module and let B the endomorphism algebra of T. We compare... Let A be a basic connected finite dimensional algebra over a field k and let Q be the ordinary quiver of A. To any presentation of A with Q and admissible relations, R. Martinez-Villa and J. de La... en.scientificcommons.org /patrick_le_meur   (1002 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. Consequently, ACF's behave in the same way as as any other field implemented in Magma; all standard algorithms implemented for generic fields and which use factorization work without change (for example, the Jordan form of a matrix). magma.maths.usyd.edu.au /magma/Features/node131.html   (305 words)

 New Simple Lie algebras: Melikyan Algebras   (Site not responding. Last check: 2007-10-12) 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 andÂ  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)

 Fields Institute - Midwest Model Theory Meeting Abstracts It is well-known that any polynomial ring over an uncountable algebraically closed field of characteristic zero admits such an embedding (van den Dries and Schmidt). Jet spaces in algebraic and complex analytic geometry are useful tools in studying the universal family of subvarieties of a given variety. A {\em Lie differential field} (LDF) is a field given with a finite number of derivations satisfying some given commutation relations. www.fields.utoronto.ca /programs/scientific/03-04/mwmt/abstracts.html   (756 words)

 [No title] 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)

 Algebraic Geometry   (Site not responding. Last check: 2007-10-12) Algebraic geometry is the study of geometric objects by means of algebra. Under the bijection of the proposition, nonempty irreducible closed algebraic sets correspond to nonirrelevant proper homogeneous prime ideals. 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: 2007-10-12) For an algebraic group G, defined over an algebraically closed field of characteristic zero, there is a natural partial order on the set of G-actions on algebraic varieties: X >= Y if there exists a dominant G-equivariant rational map (i.e., a compression) from X to Y. Alternatively, one can consider regular, rather than rational, compressions. Abstract: Let G be an algebraic group and X be an irreducible algebraic variety with a generically free G-action, all defined over an algebraically closed field of characteristic zero. In this paper we classify the smooth closed subvarieties of Alg_r which are invariant under the transport of structure action and study the singularities which may occur. www.math.ubc.ca /~reichst/abstract.html   (2805 words)

 Nimber - Wikipedia, the free encyclopedia Except for the fact that nimbers form a proper class and not a set, the class of nimbers determines an algebraically closed field of characteristic 2. The nimber square of a Fermat 2-power x is equal to 3x/2 as evaluated under the ordinary multiplication of natural numbers. The smallest algebraically closed field of nimbers is the set of nimbers less than the ordinal en.wikipedia.org /wiki/Nimber   (460 words)

 Theory of a differential equation   (Site not responding. Last check: 2007-10-12) Many mathematicians will be familiar with the idea of an algebraically closed field, a field in which every system of polynomial equations (and inequations) has all possible solutions. The idea of a differentially closed field is similar - it is a field with a differentiation operator in which every system of (polynomial) differential equations and inequations has all possible solutions. By "complete theory" I mean complete first-order theory, in the language of differential fields, or of fields together with a relation giving the solution set of the equation. www.maths.ox.ac.uk /~kirby/theory_of_de.html   (334 words)

 Xavier Vidaux   (Site not responding. Last check: 2007-10-12) We prove that the set of rational integers is positive existentially definable in the field M of meromorphic functions on K in the language L of rings augmented by a constant symbol for the independent variable z and by a symbol for the unary relation "the function x takes the value 0 at 0". Abstract : Let K and K' be two elliptic fields with complex multiplication over an algebraically closed field k of characteristic 0, non k-isomorphic, and let C and C' be two curves with respectively K and K' as function fields. This theorem is an analogue of a theorem from David A. Pierce in the language of k-algebras. www.maths.ox.ac.uk /~vidaux/indexEnglish.html   (503 words)

 Absolute Field One may construct an absolute field isomorphic to the current subfield represented by an algebraically closed field. The construction of the absolute field may be very expensive, as it involves factoring polynomials over successive subfields. In fact, it is often the case that the degree of the absolute field is an extremely large integer, so that an absolute field is not practically representable, yet the system may allow one to compute effectively with the original non-absolute presentation. www.math.wayne.edu /answers/magma2.10/htmlhelp/text757.htm   (495 words)

 Closed Field And a proper subfield misses certain algebraic elements, and is not closed. If f is not closed, pick an element u not in f and let c map u to a root of the corresponding polynomial over e. Other countable fields, such as quotients of polynomials with integer coefficients, can be closed without resorting to choice, and the closure is unique up to isomorphism. www.mathreference.com /fld,closed.html   (687 words)

 Algebraic extension   (Site not responding. Last check: 2007-10-12) If F and G are fields and G contains F, then the field extension G/F is called algebraic if every element of G is algebraic over F, meaning that for every element x of G there exists a non-zero polynomial p with coefficients in F such that p(x) = 0. 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. 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 brandt.kurowski.net /projects/lsa/wiki/view.cgi?doc=504   (268 words)

 Project-Team - Spaces These values are searched in an algebraically closed field containing K or in the field of the reals if K is a real field (The problem of finding the integer or the rational solutions is known to be undecidable.). When K is the finite field with q elements, which is usually the case in cryptography, one may prescribe that all the solutions are in K by adding the equation x In this case, the set of solutions does not depend on the algebraically closed field which is chosen. www.inria.fr /rapportsactivite/RA2003/spaces/module4.html   (276 words)

 redlog User Manual - Contexts   (Site not responding. Last check: 2007-10-12) The quantifier elimination actually requires the more restricted class of real closed fields, while most of the tool-like algorithms are generally correct for ordered fields. This is for computing with formulas over classes of p-adic valued extension fields of the rationals, usually the fields of p-adic numbers for some prime p. Algebraically closed fields such as the complex numbers. www.fmi.uni-passau.de /~redlog/htmldoc/redlog_3.html   (167 words)

 First-order Model Theory For example the field of real numbers forms a structure R whose elements are the real numbers, with signature consisting of the individual constant 0 to name the number zero, a 1-ary function symbol - for minus, and two 2-ary function symbols + and. The programme is broadly to classify structures according to (a) what groups or fields are interpretable in them (in the sense sketched in the entry on model theory) and (b) whether or not the structures have ‘modular geometries’; and then to use this classification to solve problems in model theory and geometry. Alex Wilkie showed that the field of real numbers with a symbol for exponentiation is o-minimal and has a model-complete complete theory, and thereby gave a positive answer to Tarski's old problem of whether this structure allows a quantifier elimination (though his method was very far from the syntactic analysis that Tarski had in mind). plato.stanford.edu /entries/modeltheory-fo   (6179 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)

 [No title] Prehomogeneous spaces over algebraically closed fields of characteristic zero were classified by Sato and Kimura in [SK]; this was extended to arbitrary characteristic by Chen in [Ch1,Ch2]. Observe that $k$ is the fixed field of $A$, and since $k$ is algebraically closed, every orbit of $A$ either has size 1 or is infinite. As usual, $K$ denotes an algebraically closed field of characteristic $p \geq 0$, and $V$ is a finite-dimensional vector space over $K$. www.amsta.leeds.ac.uk /Pure/preprints/hdm/hdm10   (9385 words)

 [No title]   (Site not responding. Last check: 2007-10-12) 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)

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