Linearly-disjoint extensions
of a field 
Two subextensions 
and 
of an extension 
of 
such that the subalgebra generated by 
and 
in 
is (isomorphic to) the tensor product 
over 
(cf. Extension of a field). Let 
and 
be arbitrary subrings of an extension 
of 
, containing 
, and let 
be the subring of 
generated by 
and 
. There is always a ring homomorphism 
that associates with an element 
, 
, 
, the product 
in 
. The algebras 
and 
are said to be linearly disjoint over 
if 
is an isomorphism of 
onto 
. In this case, 
. For 
and 
to be linearly disjoint over 
it is sufficient that there is a basis of 
over 
that is independent over 
. If 
is a finite extension of 
, then the degree of the extension 
does not exceed the degree of extension 
and equality holds if and only if 
and 
are linearly disjoint.
References
| [1] | N. Bourbaki, "Algebra" , Elements of mathematics , 1 , Springer (1988) pp. Chapts. 4–7 (Translated from French) |
Comments
References
| [a1] | O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975) |
Linearly-disjoint extensions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linearly-disjoint_extensions&oldid=21153