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 ?
I aim to calculate the minimal SNC model of more general hyperelliptic curves, those with tame potentially semistable reduction, also via cluster pictures - beginning with curves over DVRs with algebraically closed residue fields, and hopefully moving onto a general DVR.
He does assume they have already seen some non-archimedean fields, usually the p-adic numbers and hopefully the complete algebraically closed p-adic field Cp, but he reviews them briefly just in case.
In the following, we assume that all algebras are over an algebraically closed field F with characteristic zero, Id is the identity mapping, and Z is the set of integers.
We work over an algebraically closed field of arbitrary characteristic throughout this paper.
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.
Poonen, Isomorphism types of commutative algebras of finite rank over an algebraically closed field, K.
Throughout this paper, k is an algebraically closed field, A is a finite dimensional k-algebra.
of Newfoundland, Canada) introduce theory of gradings on Lie algebras, with a focus on classifying gradings on simple finite-dimensional Lie algebras over algebraically closed fields.
Now let K denote an algebraically closed field of characteristic 0 with K[x] the corresponding polynomial ring and
The equation (24) admits certainly a solution, in the body C algebraically closed. Consequently, the number p will be complex.
Let n be a positive integer, and let V be a 2n-dimensional vector space over an algebraically closed field K of characteristic 0.
Thus we may assume that R is centrally closed over C which is either finite or algebraically closed and f(x, y) = 0 for all x, y [member of] R.