algebraically closed field


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algebraically closed field

[¦al·jə¦brā·ik·lē ¦klōzd ′fēld]
(mathematics)
A field F such that every polynomial of degree equal to or greater than 1 with coefficients in F has a root in F.
A field F is said to be algebraically closed in an extension field K if any root in K of a polynominal with coefficients in F also lies in F. Also known as algebraically complete field.
References in periodicals archive ?
Let H be a semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is commutative if and only if [z.sub.2] [member of] k.
Let n be a positive integer, and let V be a 2n-dimensional vector space over an algebraically closed field K of characteristic 0, with a fixed basis [e.sub.1], [e.sub.2], ..., [e.sub.n], [e.sub.1]*, [e.sub.2]*, ..., [e.sub.n]*.
A Lie algebra G over an algebraically closed field of characteristic 0 is said to be Frobenius if there exists a linear form f [member of] [G.sup.*] such that the bilinear form [[PHI].sub.f] f on G is nondegenerate.
In this paper we shall consider a moduli space [M.sub.G] of stable principal G-bundles over a smooth projective variety X of dimension n, defined over an algebraically closed field k of characteristic 0, and we shall describe a natural procedure which leads to the construction of closed differential forms on [M.sub.G] starting with some cohomology classes on X.
Our approach was suggested by a reexamination of the trace map Tr for projective curves X over algebraically closed fields: Note that for n invertible on X, Tr is the composition:
4.2 hold true over any algebraically closed field k.
Let k be an algebraically closed field of characteristic 0, let A be a reduced Noetherian k-algebra, and let (M, [nabla]) be a finitely generated torsion free A-module of rank one with a (not necessarily integrable) connection.
This graduate textbook introduces the theory of error-correcting codes, working over an arbitrary finite field; algebraic curves, working over an algebraically closed field; geometry over a finite field; and constructions of algebraic geometry codes.
Keywords abc-theorem; abc-conjecture; algebraically closed field; Wronskian; Diophantine equations.
Let [G.sub.0] be a one-parameter formal group law of height 2 over the algebraically closed field k of characteristic p.
Let k be an algebraically closed field of characteristic zero, which is complete for a non-trivial non-Archimedean absolute value [absolute value of *].
A numerical Godeaux surface is a minimal surface X of general type over an algebraically closed field with [K.sup.2.sub.X] = 1 and [chi][O.sub.X] = 1.