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In algebra, a field ''k'' is said to be perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has distinct roots. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k'' is separable. * Every algebraic extension of ''k'' is separable. * Either ''k'' has characteristic 0, or, when ''k'' has characteristic ''p'' > 0, every element of ''k'' is a ''p''th power. * Either ''k'' has characteristic 0, or, when ''k'' has characteristic ''p'' > 0, the Frobenius endomorphism ''x''→''x''''p'' is an automorphism of ''k'' * The separable closure of ''k'' is algebraically closed. * Every reduced commutative ''k''-algebra ''A'' is a separable algebra; i.e., is reduced for every field extension ''F''/''k''. (see below) Otherwise, ''k'' is called imperfect. In particular, all fields of characteristic zero and all finite fields are perfect. Perfect fields are significant because Galois theory over these fields becomes simpler, since the general Galois assumption of field extensions being separable is automatically satisfied over these fields (see third condition above). More generally, a ring of characteristic ''p'' (''p'' a prime) is called perfect if the Frobenius endomorphism is an automorphism.〔, Section II.4〕 (This is equivalent to the above condition "every element of ''k'' is a ''p''th power" for integral domains.) ==Examples== Examples of perfect fields are: * every field of characteristic zero, e.g. the field of rational numbers or the field of complex numbers; * every finite field, e.g. the field F''p'' = Z/''p''Z where ''p'' is a prime number; * every algebraically closed field; * the union of a set of perfect fields totally ordered by extension; * fields algebraic over a perfect field. In fact, most fields that appear in practice are perfect. The imperfect case arises mainly in algebraic geometry in characteristic ''p''>0. Every imperfect field is necessarily transcendental over its prime subfield (the minimal subfield), because the latter is perfect. An example of an imperfect field is *the field of all rational functions in an indeterminate , where ''k'' has characteristic ''p''>0 (because ''X'' has no ''p''-th root in ''k''(''X'')). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Perfect field」の詳細全文を読む スポンサード リンク
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