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 ?
In a three-volume work, Strade classifies the simple Lie algebras over an algebraically closed field of characteristic p equal to or larger than 3, in the sense that he presents a list of simple Lie algebras and a proof that this list is complete.
Throughout H is a d-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0 and H* is its dual which is a semisimple Hopf algebra as well.
Let K be an algebraically closed field with a complete and non-Archimedean norm [absolute value of (*)].
Throughout this paper, k is an algebraically closed field, A is a finite dimensional k-algebra.
Now let K denote an algebraically closed field of characteristic 0 with K[x] the corresponding polynomial ring and
Let n be a positive integer, and let V be a 2n-dimensional vector space over an algebraically closed field K of characteristic 0.
Throughout this paper K is an algebraically closed field of characteristic 0.
When X is a non-singular projective variety defined over an algebraically closed field k of characteristic 0 and G is a connected reductive algebraic group over k, moduli spaces of (semi)stable principal G-bundles over X are known to exist and to be quasi-projective schemes (usually singular).
Let F be a fixed algebraically closed field of characteristic 0.
In this section we explain how the trace map on the second cohomology of a curve over an algebraically closed field k can be obtained as a special case of the construction in section 1.
We work over an algebraically closed field k of characteristic p [not equal to] 2.
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.