Editing Algebraically closed field (section)

==Other properties==
If ''F'' is an algebraically closed field and ''n'' is a natural number, then ''F'' contains all ''n''th roots of unity, because these are (by definition) the ''n'' (not necessarily distinct) zeroes of the polynomial ''x<sup>n</sup>''&nbsp;&minus;&nbsp;1. A field extension that is contained in an extension generated by the roots of unity is a ''cyclotomic extension'', and the extension of a field generated by all roots of unity is sometimes called its ''cyclotomic closure''. Thus algebraically closed fields are cyclotomically closed. The converse is not true. Even assuming that every polynomial of the form ''x<sup>n</sup>''&nbsp;&minus;&nbsp;''a'' splits into linear factors is not enough to assure that the field is algebraically closed.

If a proposition which can be expressed in the language of [[first-order logic]] is true for an algebraically closed field, then it is true for every algebraically closed field with the same [[Characteristic (algebra)|characteristic]]. Furthermore, if such a proposition is valid for an algebraically closed field with characteristic&nbsp;0, then not only is it valid for all other algebraically closed fields with characteristic&nbsp;0, but there is some natural number ''N'' such that the proposition is valid for every algebraically closed field with characteristic&nbsp;''p'' when ''p''&nbsp;&gt;&nbsp;''N''.<ref>See subsections ''Rings and fields'' and ''Properties of mathematical theories'' in §2 of J. Barwise's "An introduction to first-order logic".</ref>

Every field ''F'' has some extension which is algebraically closed. Such an extension is called an '''algebraically closed extension'''. Among all such extensions there is one and only one ([[Up to|up to isomorphism]], but not [[essentially unique|unique isomorphism]]) which is an [[algebraic extension]] of ''F'';<ref>See Lang's ''Algebra'', §VII.2 or van der Waerden's ''Algebra I'', §10.1.</ref> it is called the [[algebraic closure]] of ''F''.

The theory of algebraically closed fields has [[quantifier elimination]].