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In abstract algebra, the transcendence degree of a field extension ''L'' /''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ''L'' over ''K''. A subset ''S'' of ''L'' is a transcendence basis of ''L'' /''K'' if it is algebraically independent over ''K'' and if furthermore ''L'' is an algebraic extension of the field ''K''(''S'') (the field obtained by adjoining the elements of ''S'' to ''K''). One can show that every field extension has a transcendence basis, and that all transcendence bases have the same cardinality; this cardinality is equal to the transcendence degree of the extension and is denoted trdeg''K'' ''L'' or trdeg(''L'' /''K''). If no field ''K'' is specified, the transcendence degree of a field ''L'' is its degree relative to the prime field of the same characteristic, i.e., Q if ''L'' is of characteristic 0 and F''p'' if ''L'' is of characteristic ''p''. The field extension ''L'' /''K'' is purely transcendental if there is a subset ''S'' of ''L'' that is algebraically independent over ''K'' and such that ''L'' = ''K''(''S''). == Examples == *An extension is algebraic if and only if its transcendence degree is 0; the empty set serves as a transcendence basis here. *The field of rational functions in ''n'' variables ''K''(''x''1,...,''x''''n'') is a purely transcendental extension with transcendence degree ''n'' over ''K''; we can for example take as a transcendence base. *More generally, the transcendence degree of the function field ''L'' of an ''n''-dimensional algebraic variety over a ground field ''K'' is ''n''. *Q(√2, π) has transcendence degree 1 over Q because √2 is algebraic while π is transcendental. *The transcendence degree of C or R over Q is the cardinality of the continuum. (This follows since any element has only countably many algebraic elements over it in Q, since Q is itself countable.) *The transcendence degree of Q(π, ''e'') over Q is either 1 or 2; the precise answer is unknown because it is not known whether π and ''e'' are algebraically independent. 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「Transcendence degree」の詳細全文を読む スポンサード リンク
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