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. On the other hand, 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).
Contents
- Compositum of fields
- The tensor product as ring
- Analysis of the ring structure
- Examples
- Classical theory of real and complex embeddings
- Consequences for Galois theory
- References
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
Firstly, 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
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,
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.
The tensor product as ring
To get a general theory, one needs to consider a ring structure on
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
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
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
Examples
For example, if K is generated over ℚ by the cube root of 2, then
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) = Xp − Twith 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(T1/p) (the splitting field of P) then L/K is an example of a purely inseparable field extension. In
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 (implicitly, often) a basic tool. If K is an extension of ℚ of finite degree n,
This idea applies also to
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.