|
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field that contains the base field and satisfies additional properties. For instance, the set Q(√2) = is the smallest extension of Q that includes every real solution to the equation ''x''2 = 2. == Definitions == Let ''L'' be a field. A subfield of ''L'' is a subset ''K'' of ''L'' that is closed under the field operations of ''L'' and under taking inverses in ''L''. In other words, ''K'' is a field with respect to the field operations inherited from ''L''. The larger field ''L'' is then said to be an extension field of ''K''. To simplify notation and terminology, one says that ''L'' / ''K'' (read as "''L'' over ''K''") is a field extension to signify that ''L'' is an extension field of ''K''. If ''L'' is an extension of ''F'' which is in turn an extension of ''K'', then ''F'' is said to be an intermediate field (or intermediate extension or subextension) of the field extension ''L'' / ''K''. Given a field extension ''L'' / ''K'' and a subset ''S'' of ''L'', the smallest subfield of ''L'' which contains ''K'' and ''S'' is denoted by ''K''(''S'')—i.e. ''K''(''S'') is the field generated by adjoining the elements of ''S'' to ''K''. If ''S'' consists of only one element ''s'', ''K''(''s'') is a shorthand for ''K''(). A field extension of the form ''L'' = ''K''(''s'') is called a simple extension and ''s'' is called a primitive element of the extension. Given a field extension ''L'' / ''K'', the larger field ''L'' can be considered as a vector space over ''K''. The elements of ''L'' are the "vectors" and the elements of ''K'' are the "scalars", with vector addition and scalar multiplication obtained from the corresponding field operations. The dimension of this vector space is called the degree of the extension and is denoted by (). An extension of degree 1 (that is, one where ''L'' is equal to ''K'') is called a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions, respectively. Depending on whether the degree is finite or infinite the extension is called a finite extension or infinite extension. 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「field extension」の詳細全文を読む スポンサード リンク
|