First, one defines the notion of the compositum of fields.
This construction occurs frequently in field theory.
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
[1] 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,
A significant case in the theory of cyclotomic fields is that for the nth roots of unity, for n a composite number, the subfields generated by the pk th roots of unity for prime powers dividing n are linearly disjoint for distinct p.[2] To get a general theory, one needs to consider a ring structure on
(see Tensor product of algebras).
This formula is multilinear over N in each variable; and so defines a ring structure on the tensor product, making
into a commutative N-algebra, called the tensor product of fields.
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 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 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).
as a direct product of finitely many fields.
Furthermore this algebra is isomorphic to a direct sum of fields via the map induced by
When one performs the tensor product over this better candidate for the largest common subfield we actually get a (rather trivial) field For another example, if K is generated over
is the sum of (a copy of) K, and a splitting field of of degree 6 over
One can prove this by calculating the dimension of the tensor product over
as 9, and observing that the splitting field does contain two (indeed three) copies of K, and is the compositum of two of them.
An example leading to a non-zero nilpotent: let with K the field of rational functions in the indeterminate T over the finite field with p elements (see Separable polynomial: the point here is that P is not separable).
the element is nilpotent: by taking its pth power one gets 0 by using K-linearity.
In algebraic number theory, tensor products of fields are (implicitly, often) a basic tool.
The field factors are in 1–1 correspondence with the real embeddings, and pairs of complex conjugate embeddings, described in the classical literature.
p, in 1–1 correspondence with the completions of K for extensions of the p-adic metric on
This gives a general picture, and indeed a way of developing Galois theory (along lines exploited in Grothendieck's Galois theory).
It can be shown that for separable extensions the radical is always {0}; therefore the Galois theory case is the semisimple one, of products of fields alone.