In abstract algebra, the theory offields lacks a direct product: the direct product of two fields, considered as a ring, is never itself a field. Nonetheless, it is often required to "join" two fields K and L, either in cases where K and L are given as subfields of a larger field M, or when K and L are both field extensions of a smaller field N. The tensor product of fields is the best available construction on fields with which to discuss all the phenomena arising. As a ring, it is sometimes a field, and often a direct product of fields; it can, though, contain non-zero nilpotents. If K and L do not have isomorphic prime fields, or in other words they have different characteristics, they have no possibility of being common subfields of a field M. Correspondingly their tensor product will in that case be the trivial ring.
Compositum of fields
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. Note that this assumes some field containing both K and L. Either one starts in a situation where such a common over-field is easy to identify ; or one proves a result that allows one to place both K and L in some large enough field. In many cases one can identify K.L as a vector space tensor product, taken over the fieldN that is the intersection of K and L. For example, if one adjoins √2 to the rational field ℚ to get K, and √3 to get L, it is true that the field M obtained as K.L inside the complex numbers ℂ is as a vector space over ℚ. Subfields K and L of M are linearly disjoint 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, injective is equivalent here to bijective. Hence, when K and L are linearly disjoint finite-degree extension fields over N,, as with the aforementioned extensions of the rationals. 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 pkth roots of unity for prime powers dividing n are linearly disjoint for distinct p.
To get a general theory, one needs to consider a ring structure on. One can define the product to be . 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. 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. 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 and so provides embeddings of K and L in some field as extensions of N. In this way one can analyse the structure of : there may in principle be a non-zero nilradical - and after taking the quotient by that one can speak of the product of all embeddings of K and L in various M, overN. 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. 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 field embeddings for K and L in some extension M.
Examples
For example, if K is generated over ℚ by the cube root of 2, then is the product of K, and a splitting field of of degree 6 over ℚ. One can prove this by calculating the dimension of the tensor product over ℚ as 9, and observing that the splitting field does contain two copies of K, and is the compositum of two of them. That incidentally shows that R = in this case. 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.. If L is the field extension K then L/K is an example of a purely inseparable field extension. In the element is nilpotent: by taking its pth power one gets 0 by using K-linearity.
Classical theory of real and complex embeddings
In algebraic number theory, tensor products of fields are a basic tool. If K is an extension of ℚ of finite degree n, is always a product of fields isomorphic to ℝ or ℂ. The totally real number fields are those for which only real fields occur: in general there are r1 real and r2 complex fields, with r1 + 2r2 = n as one sees by counting dimensions. The field factors are in 1–1 correspondence with the real embeddings, and pairs of complex conjugate embeddings, described in the classical literature. This idea applies also to where ℚp is the field of p-adic numbers. This is a product of finite extensions of ℚp, in 1–1 correspondence with the completions of K for extensions of the p-adic metric on ℚ.
This gives a general picture, and indeed a way of developing Galois theory . It can be shown that for separable extensions the radical is always ; therefore the Galois theory case is the semisimple one, of products of fields alone.
