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In field theory, a simple extension is a field extension which is generated by the adjunction of a single element. Simple extensions are well understood and can be completely classified. The primitive element theorem provides a characterization of the finite simple extensions. == Definition == A field extension ''L''/''K'' is called a simple extension if there exists an element θ in ''L'' with : The element θ is called a primitive element, or generating element, for the extension; we also say that ''L'' is generated over ''K'' by θ. Every finite field is a simple extension of the prime field of the same characteristic. More precisely, if ''p'' is a prime number and the field of ''q'' elements is a simple extension of degree ''d'' of This means that it is generated by an element θ which is a root of an irreducible polynomial of degree ''d''. However, in this case, θ is normally not referred to as a ''primitive element''. In fact, a primitive element of a finite field is usually defined as a generator of the field's multiplicative group. More precisely, by little Fermat theorem, the nonzero elements of (i.e. its multiplicative group) are the roots of the equation : that is the (''q''-1)-th roots of unity. Therefore, in this context, a primitive element is a primitive (''q''-1)-th root of unity, that is a generator of the multiplicative group of the nonzero elements of the field. Clearly, a group primitive element is a field primitive element, but the contrary is false. Thus the general definition requires that every element of the field may be expressed as a polynomial in the generator, while, in the realm of finite fields, every nonzero element of the field is a pure power of the primitive element. To distinguish these meanings one may use field primitive element of L over K for the general notion, and group primitive element for the finite field notion. 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「simple extension」の詳細全文を読む スポンサード リンク
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