In
mathematics, the tensor product of two
fields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
*Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...
is their
tensor product as
algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
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
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive id ...
.
The tensor product of two fields is sometimes a field, and often a
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of fields; In some cases, it can contain non-zero
nilpotent element
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
s.
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''. 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 to get ''K'', and √3 to get ''L'', it is true that the field ''M'' obtained as ''K''.''L'' inside the complex numbers
is (
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
isomorphism)
:
as a vector space over
. (This type of result can be verified, in general, by using the
ramification theory of
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
.)
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, 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 field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of th ...
s 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
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
structure on
. One can define the product
to be
(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
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
into ''M'' defined by:
:
The
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learni ...
of γ will be a
prime ideal of the tensor product; and
conversely
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposit ...
any prime ideal of the tensor product will give a homomorphism of ''N''-algebras to an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
(inside a
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
) 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
: 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 In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are ...
). 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 (essentially distinct) field embeddings for ''K'' and ''L'' in some extension ''M''.
Examples
For example, if ''K'' is generated over
by the
cube root
In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. ...
of 2, then
is the product of (a copy of) ''K'', and a
splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors.
Definition
A splitting field of a polyn ...
of
:''X''
  3 − 2,
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 (indeed three) 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
:''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
In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors.
Definition
A splitting field of a polyn ...
of ''P'') then ''L''/''K'' is an example of a
purely inseparable field extension In algebra, a purely inseparable extension of fields is an extension ''k'' ⊆ ''K'' of fields of characteristic ''p'' > 0 such that every element of ''K'' is a root of an equation of the form ''x'q'' = ''a'', wit ...
. 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
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
, tensor products of fields are (implicitly, often) 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
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
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
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
(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.
See also
*
Extension of scalars—tensor product of a field extension and a vector space over that field
Notes
References
*
*
*
*
*{{Cite book , last1=Zariski , first1=Oscar , author1-link=Oscar Zariski , last2=Samuel , first2=Pierre , author2-link=Pierre Samuel , title=Commutative algebra I , orig-year=1958 , publisher=
Springer-Verlag , series=Graduate Texts in Mathematics , isbn=978-0-387-90089-6 , mr=0090581 , year=1975 , volume=28
External links
MathOverflow thread on the definition of linear disjointness
Field (mathematics)