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Concept
Tensor product of fields
Summary
In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subfield.
The tensor product of two fields is sometimes a field, and often a direct product of fields; In some cases, it can contain non-zero nilpotent elements.
The tensor product of two fields expresses in a single structure the different way to embed the two fields in a common extension field.
First, 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 K.L, is defined to be 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 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 to get K, and √3 to get L, it is true that the field M obtained as K.L inside the complex numbers is (up to isomorphism)
as a vector space over . (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, injectivity is equivalent here to bijectivity.
Official source
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