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In field theory, the primitive element theorem or Artin's theorem on primitive elements is a result characterizing the finite degree field extensions that possess a primitive element, or simple extensions. It says that a finite extension is simple if and only if there are only finitely many intermediate fields. In particular, finite separable extensions are simple. == Terminology == Let be a finite field extension. An element is said to be a ''primitive element'' for when : In this situation, the extension is referred to as a ''simple extension''. Then every element ''x'' of ''E'' can be written in the form : where for all ''i'', and is fixed. That is, if is separable of degree ''n'', there exists such that the set : is a basis for ''E'' as a vector space over ''F''. For instance, the extensions and are simple extensions with primitive elements and ''x'', respectively ( denotes the field of rational functions in the indeterminate ''x'' over ). 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「primitive element theorem」の詳細全文を読む スポンサード リンク
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