Algebraically closed field

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In mathematics, a field F is algebraically closed if every non-constant polynomial in F[x] (the univariate polynomial ring with coefficients in F) has a root in F. In other words, a field is algebraically closed if the fundamental theorem of algebra holds for it.

Every field ${\displaystyle K}$ is contained in an algebraically closed field ${\displaystyle C,}$ and the roots in ${\displaystyle C}$ of the polynomials with coefficients in ${\displaystyle K}$ form an algebraically closed field called an algebraic closure of ${\displaystyle K.}$ Given two algebraic closures of ${\displaystyle K}$ there are isomorphisms between them that fix the elements of ${\displaystyle K.}$