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. Note that the construction here assumes the common subfield N; but does not assume 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 into M defined by:
The kernel of γ will be a prime ideal of the tensor product; and 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 : there may in principle be a non-zero Jacobson radical (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 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 of M.
Read more about this topic: Tensor Product Of Fields
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