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An extension field $L$ of $K$ is an \emph{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.
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An extension field $L$ of $K$ is an \emph{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 a field are isomorphic as fields, but not necessarily canonically isomorphic.
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