Algebraically closed field

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In mathematics, a field F is algebraically closed if every non-constant polynomial 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. For example, the field of real numbers is not algebraically complete because the polynomial ${\displaystyle x^{2}+1}$ has no real roots, while the field of complex numbers is algebraically closed.

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 ${\displaystyle K.}$

Algebraically closed fields appear in the following chain of class inclusions:

rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ euclidean domains ⊃ fields ⊃ algebraically closed fields