Editing Tensor product of fields (section)

==Compositum of fields==

First, one defines the notion of the compositum of fields. This construction occurs frequently in [[field theory (mathematics)|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 ''K.L'', is defined to be <math> K.L = k(K \cup L) </math> where the right-hand side denotes the extension generated by ''K'' and ''L''. This assumes ''some'' field containing both ''K'' and ''L''. Either one starts in a situation where an ambient field is easy to identify (for example if ''K'' and ''L'' are both subfields of the [[complex number]]s), 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 numbers|rational field]] <math>\mathbb{Q}</math> to get ''K'', and √3 to get ''L'', it is true that the field ''M'' obtained as ''K''.''L'' inside the complex numbers <math>\mathbb{C}</math> is ([[up to]] isomorphism)

:<math>K\otimes_{\mathbb Q}L</math>

as a vector space over <math>\mathbb{Q}</math>. (This type of result can be verified, in general, by using the [[ramification (mathematics)|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

:<math>K\otimes_NL</math>

to ''K''.''L'' is [[injective]].<ref>{{springer|id=L/l059560|title=Linearly-disjoint extensions}}</ref> Naturally enough this isn't always the case, for example when ''K'' = ''L''. When the degrees are finite, injectivity is equivalent here to [[bijectivity]]. Hence, when ''K'' and ''L'' are linearly disjoint finite-degree extension fields over ''N'', <math>K.L \cong K \otimes_N L</math>, as with the aforementioned extensions of the rationals.

A significant case in the theory of [[cyclotomic field]]s is that for the ''n''th [[roots of unity]], for ''n'' a [[composite number]], the subfields generated by the ''p''<sup>''k''</sup>th roots of unity for [[prime power]]s dividing ''n'' are linearly disjoint for distinct ''p''.<ref>{{springer|id=c/c027570|title=Cyclotomic field}}</ref>