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# algebraically closed

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.

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