The **tensor product of fields** is the best available operation on fields with which to discuss the phenomena. As a ring, it is sometimes a field, and often a direct product of fields; it can though contain non-zero nilpotents (see radical of a ring).

Table of contents |

2 The tensor product as ring 3 Analysis of the ring structure 4 Examples 5 Consequences for Galois theory |

Firstly, in field theory, the **compositum** of subfields K and L of a field M is defined, without a problem, as the smallest subfield of M containing both K and L. It can be written K.L. In many cases we 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 we adjoin to the rational field **Q** the square root of 2 to get K, and the square root of 3 to get L, it is true that the field M obtained as K.L inside the complex numbers **C** is K_{Q}L as a vector space over **Q**. This kind of result can be proved in general using the ramification theory of algebraic number theory. We say that subfields K and L of M are **linearly disjoint** (over a subfield N) when in this way the natural N-linear map of K_{N}L 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.

To get a general theory, we need to consider a ring structure on K_{N}L. We can define *a 'b*

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 K_{N}L: there may in principle be a radical (intersection of all prime ideals) - and after taking the quotient by that we 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 so Artinian). We can then say that if R is the radical we have K_{N}L/R 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.

For example, if K is generated over **Q** by the cube root of 2, then K_{Q}K is the product of (a copy) of K, and a splitting field of X^{3} - 2, 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 (indeed three) copies of K, and is the compositum of two of them. That incidentally shows that R = {0} in this case.

An example leading to a non-zero nilpotent: let P(X) = X^{p} - T with K the field of rational functions in the indeterminate T over the finite field with p elements. (See separable polynomial: gthe point here is that P is *not* separable). If L is the field extension K(T^{1/p}) (the splitting field of P) then L/K is an example of a purely inseparable field extension. In L_{K}L the element T^{1/p}1 - 1T^{1/p} is nilpotent: by taking its *p*th power one gets 0 by using K-linearity.

In algebraic number theory, tensor products of fields are (implicitly, often) a basic tool. If K is an extension of **Q** of finite degree *n*, K_{Q}**R** 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 *r* real and *s* complex fields, with *r* + 2*s* = *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 K_{Q}**Q**_{p}, where **Q**_{p} is the field of p-adic numbers. This is a product of finite extensions of **Q**_{p}, 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
(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.