Editing Tensor product of fields (section)

==Analysis of the ring structure==

The structure of the ring can be analysed by considering all ways of embedding both ''K'' and ''L'' in some field extension of ''N''. The construction here assumes the common subfield ''N''; but does not assume ''[[A priori and a posteriori|a priori]]'' that ''K'' and ''L'' are subfields of some field ''M'' (thus getting round the caveats about constructing a compositum field). Whenever one embeds ''K'' and ''L'' in such a field ''M'', say using embeddings α of ''K'' and β of ''L'', there results a [[ring homomorphism]] γ from <math>K \otimes_N L</math> into ''M'' defined by:

:<math>\gamma(a\otimes b) = (\alpha(a)\otimes1)\star(1\otimes\beta(b)) = \alpha(a).\beta(b).</math>

The [[kernel (algebra)|kernel]] of γ will be a [[prime ideal]] of the tensor product; and [[converse (logic)|conversely]] any prime ideal of the tensor product will give a homomorphism of ''N''-algebras to an [[integral domain]] (inside a [[field of fractions]]) and so provides embeddings of ''K'' and ''L'' in some field as extensions of (a copy of) ''N''.

In this way one can analyse the structure of <math>K \otimes_N L</math>: there may in principle be a non-zero [[Nilradical of a ring|nilradical]] (intersection of all prime ideals) – and after taking the quotient by that one can speak of the product of all embeddings of ''K'' and ''L'' in various ''M'', ''over'' ''N''.

In case ''K'' and ''L'' are finite extensions of ''N'', the situation is particularly simple since the tensor product is of finite dimension as an ''N''-algebra (and thus an [[Artinian ring]]). One can then say that if ''R'' is the radical, one has <math>(K \otimes_N L) / R</math> as a direct product of finitely many fields. Each such field is a representative of an [[equivalence class]] of (essentially distinct) field embeddings for ''K'' and ''L'' in some extension ''M''.