In
mathematics,
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 additio ...
''A'', ''B'' over a
field ''k'' inside some
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
of ''k'' are said to be linearly disjoint over ''k'' if the following equivalent conditions are met:
*(i) The map
induced by
is
injective.
*(ii) Any ''k''-
basis of ''A'' remains
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ...
over ''B''.
*(iii) If
are ''k''-bases for ''A'', ''B'', then the products
are linearly independent over ''k''.
Note that, since every subalgebra of
is a
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Do ...
, (i) implies
is a domain (in particular
reduced). Conversely if ''A'' and ''B'' are fields and either ''A'' or ''B'' is an
algebraic extension
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field e ...
of ''k'' and
is a domain then it is a field and ''A'' and ''B'' are linearly disjoint. However, there are examples where
is a domain but ''A'' and ''B'' are not linearly disjoint: for example, ''A'' = ''B'' = ''k''(''t''), the
field of rational functions over ''k''.
One also has: ''A'', ''B'' are linearly disjoint over ''k''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
subfields of
generated by
, resp. are linearly disjoint over ''k''. (cf.
Tensor product of fields
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 sub ...
)
Suppose ''A'', ''B'' are linearly disjoint over ''k''. If
,
are subalgebras, then
and
are linearly disjoint over ''k''. Conversely, if any finitely generated subalgebras of algebras ''A'', ''B'' are linearly disjoint, then ''A'', ''B'' are linearly disjoint (since the condition involves only finite sets of elements.)
See also
*
Tensor product of fields
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 sub ...
References
* P.M. Cohn (2003). Basic algebra
Algebra
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