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In abstract algebra, the theory of fields lacks a direct product: the direct product of two fields, considered as a ring is ''never'' itself a field. On the other hand it is often required to 'join' two fields ''K'' and ''L'', either in cases where ''K'' and ''L'' are given as subfields of a larger field ''M'', or when ''K'' and ''L'' are both field extensions of a smaller field ''N'' (for example a prime field). The tensor product of fields is the best available construction on fields with which to discuss all the phenomena arising. As a ring, it is sometimes a field, and often a direct product of fields; it can, though, contain non-zero nilpotents (see radical of a ring). If ''K'' and ''L'' do not have isomorphic prime fields, or in other words they have different characteristics, they have no possibility of being common subfields of a field ''M''. Correspondingly their tensor product will in that case be the trivial ring (collapse of the construction to nothing of interest). ==Compositum of fields== Firstly, one defines the notion of the compositum of fields. This construction occurs frequently in field theory. The idea behind the compositum is to make the smallest field containing two other fields. In order to formally define the compositum, one must first specify a tower of fields. Let ''k'' be a field and ''L'' and ''K'' be two extensions of ''k''. The compositum, denoted ''KL'' is defined to be where the right-hand side denotes the extension generated by ''K'' and ''L''. Note that this assumes ''some'' field containing both ''K'' and ''L''. Either one starts in a situation where such a common over-field is easy to identify (for example if ''K'' and ''L'' are both subfields of the complex numbers); or one proves a result that allows one to place both ''K'' and ''L'' (as isomorphic copies) in some large enough field. In many cases one can identify ''K''.''L'' as a vector space tensor product, taken over the field ''N'' that is the intersection of ''K'' and ''L''. For example if one adjoins √2 to the rational field ℚ to get ''K'', and √3 to get ''L'', it is true that the field ''M'' obtained as ''K''.''L'' inside the complex numbers ℂ is (up to isomorphism) : as a vector space over ℚ. (This type of result can be verified, in general, by using the ramification theory of algebraic number theory.) Subfields ''K'' and ''L'' of ''M'' are linearly disjoint (over a subfield ''N'') when in this way the natural ''N''-linear map of : to ''K''.''L'' is injective. Naturally enough this isn't always the case, for example when ''K'' = ''L''. When the degrees are finite, injective is equivalent here to bijective. Hence, when ''K'' and ''L'' are linearly disjoint finite-degree extension fields over ''N'', , as with the aforementioned extensions of the rationals. A significant case in the theory of cyclotomic fields is that for the ''n''th roots of unity, for ''n'' a composite number, the subfields generated by the ''p''''k''th roots of unity for prime powers dividing ''n'' are linearly disjoint for distinct ''p''. 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「tensor product of fields」の詳細全文を読む スポンサード リンク
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