Tensor product of fields

In abstract algebra, the theory of fields 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" (for example a prime field).
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 (see radical of a ring).
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 (collapse of the construction to nothing of interest).
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 formula_1 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 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 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 (up to isomorphism)
as a vector space over ℚ. (This type of result can be verified, in general, by using the ramification theory of algebraic number theory.)
Subfields "K" and "L" of "M" are linearly disjoint (over a subfield "N") 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", formula_4, as with the aforementioned extensions of the rationals.
A significant case in the theory of cyclotomic fields is that for the "n"th roots of unity, for "n" a composite number, the subfields generated by the "p""k"th 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 formula_5. One can define the product formula_6 to be formula_7 (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 formula_5 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". 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 formula_5 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 formula_5: 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). One can then say that if "R" is the radical, one has formula_12 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 "M".
Examples.
For example, if "K" is generated over ℚ by the cube root of 2, then formula_13 is the product of (a copy 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 (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
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). 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 formula_14 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 (implicitly, often) a basic tool. If "K" is an extension of ℚ of finite degree "n", formula_16 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"1 real and "r"2 complex fields, with "r"1 + 2"r"2 = "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 formula_17 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.
Consequences for Galois theory.
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.