Editing Algebraically closed field (section)

{{Short description|Algebraic structure where all polynomials have roots}}
{{Use dmy dates|date=October 2020}}
{{inline citations|date=September 2021}}
In [[mathematics]], a [[field (mathematics)|field]] {{math|''F''}} is '''algebraically closed''' if every [[Degree of a polynomial|non-constant polynomial]] in {{math|''F''[''x'']}} (the univariate [[polynomial ring]] with coefficients in {{math|''F''}}) has a [[Zero of a function|root]] in {{math|''F''}}. In other words, a field is algebraically closed if the [[fundamental theorem of algebra]] holds for it. 

Every field <math>K</math> is contained in an algebraically closed field <math>C,</math> and the roots in <math>C</math> of the polynomials with coefficients in <math>K</math> form an algebraically closed field called an [[algebraic closure]] of <math>K.</math> Given two algebraic closures of <math>K</math> there are isomorphisms between them that fix the elements of <math>K.</math>

Algebraically closed fields appear in the following chain of [[subclass (set theory)|class inclusions]]:
{{Commutative ring classes}}