A field $K$ is algebraically closed if every non-constant polynomial in $K[X]$ has a root in $K$

An extension field $L$ of $K$ is an algebraic closure of $K$ if $L$ is algebraically closed and every element of $L$ is algebraic over $K$ Using the axiom of choice, one can show that any field has an algebraic closure. Moreover, any two algebraic closures of a field are isomorphic as fields, but not necessarily canonically isomorphic.