Tensor product of fields should not be confused with Tensor field.
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.
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
K.L=k(K\cupL)
Q
C
K ⊗ QL
as a vector space over
Q
Subfields K and L of M are linearly disjoint (over a subfield N) when in this way the natural N-linear map of
K ⊗ NL
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,
K.L\congK ⊗ NL
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.
To get a general theory, one needs to consider a ring structure on
K ⊗ NL
(a ⊗ b)(c ⊗ d)
ac ⊗ bd
K ⊗ NL
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
K ⊗ NL
\gamma(a ⊗ b)=(\alpha(a) ⊗ 1)\star(1 ⊗ \beta(b))=\alpha(a).\beta(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 ⊗ NL
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
(K ⊗ NL)/R
To give an explicit example consider the fields
K=Q[x]/(x2-2)
L=Q[y]/(y2-2)
Q(\sqrt{2})\congK\congL ≠ K
N=Q
K ⊗ L=K ⊗ N
2-2) ⊗ | |
L=Q[x]/(x | |
Q |
Q[y]/(y2-2)\congQ(\sqrt{2})[z]/(z2-2)\congQ(\sqrt{2}) ⊗ QQ(\sqrt{2})
is not a field, but a 4-dimensional
Q
K ⊗ L\congQ(\sqrt{2})[z]/(z2-2)\congQ(\sqrt{2}) ⊕ Q(\sqrt{2})
via the map induced by
1\mapsto(1,1),z\mapsto(\sqrt{2},-\sqrt{2})
\tilde{N}=Q(\sqrt{2})
\tilde{N}=Q(\sqrt{2})\congK\congL
K ⊗ \tilde{N
For another example, if K is generated over
Q
K ⊗ QK
X  3 − 2,
of degree 6 over
Q
Q
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: 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
L ⊗ KL
T1/p ⊗ 1-1 ⊗ T1/p
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. If K is an extension of
Q
K ⊗ QR
R
C
This idea applies also to
K ⊗ QQp,
Q
Q
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 ; therefore the Galois theory case is the semisimple one, of products of fields alone.