Algebraically closed field

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> F is algebraically closed if every <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">non-constant polynomial</a> in F[x] (the univariate <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a> with coefficients in F) has a <a href="/facts/Zero_of_a_function/mlK3dhoT">root</a> in F. In other words, a field is algebraically closed if the <a href="/facts/Fundamental_theorem_of_algebra/vspflSsM">fundamental theorem of algebra</a> holds for it. 
Every field 
 
 
 
 K
 
 
 {\displaystyle K}
 
 is contained in an algebraically closed field 
 
 
 
 C
 ,
 
 
 {\displaystyle C,}
 
 and the roots in 
 
 
 
 C
 
 
 {\displaystyle C}
 
 of the polynomials with coefficients in 
 
 
 
 K
 
 
 {\displaystyle K}
 
 form an algebraically closed field called an <a href="/facts/Algebraic_closure/yjewG1mM">algebraic closure</a> of 
 
 
 
 K
 .
 
 
 {\displaystyle K.}
 
 Given two algebraic closures of 
 
 
 
 K
 
 
 {\displaystyle K}
 
 there are isomorphisms between them that fix the elements of 
 
 
 
 K
 .
 
 
 {\displaystyle K.}

Algebraically closed fields appear in the following chain of <a href="/facts/Subclass_(set_theory)/FjTF1nAr">class inclusions</a>:

<a href="/facts/Rng_(algebra)/SquNCXKe">rngs</a> ⊃ <a href="/facts/Ring_(mathematics)/JnCUXz63">rings</a> ⊃ <a href="/facts/Commutative_ring/QS1r2ovR">commutative rings</a> ⊃ <a href="/facts/Integral_domain/cylt6zxW">integral domains</a> ⊃ <a href="/facts/Integrally_closed_domain/L3kgG2q6">integrally closed domains</a> ⊃ <a href="/facts/GCD_domain/PWrf9Tkm">GCD domains</a> ⊃ <a href="/facts/Unique_factorization_domain/aYsX2swC">unique factorization domains</a> ⊃ <a href="/facts/Principal_ideal_domain/JCwAUnyW">principal ideal domains</a> ⊃ <a href="/facts/Euclidean_domain/ydwCD0fz">euclidean domains</a> ⊃ <a href="/facts/Field_(mathematics)/xAjAS4ko">fields</a> ⊃ algebraically closed fields

Algebraically closed field open-in-new

Algebraically closed field