Tensor product of fields

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.

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. This assumes some field containing both K and L. Either one starts in a situation where an ambient 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 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
as a vector space over.
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, injectivity is equivalent here to bijectivity. 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 p^kth roots of unity for prime powers dividing n are linearly disjoint for distinct p.

The tensor product as ring

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.

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 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, 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. 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

To give an explicit example consider the fields and. Clearly are isomorphic but technically unequal fields with their intersection being the prime field. Their tensor product
is not a field, but a 4-dimensional -algebra. Furthermore this algebra is isomorphic to a direct sum of fields
via the map induced by.
Morally should be considered the largest common subfield up to isomorphism of K and L via the isomorphisms. When one performs the tensor product over this better candidate for the largest common subfield we actually get a field
For another example, if K is generated over by the cube root of 2, then is the sum 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 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 p''th 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 of finite degree n, is always a product of fields isomorphic to or. 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 _p is the field of p-adic numbers. This is a product of finite extensions of _p, in 1–1 correspondence with the completions of K for extensions of the p-adic metric on.

Consequences for Galois theory

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.