Linear disjointness
In abstract algebra , two intermediate fields and a field extension are called linearly disjoint if every set of elements of that is over linearly independent is also over linearly independent. An equivalent characterization is: The figure
is injective (see tensor product for notation ). This description also shows immediately that linear disjointness is a symmetric property of and .
The intersection of linearly disjoint partial extensions is always the basic body , i.e. H.
The converse is not generally true, but at least if one of the two extensions and is finite and Galois .
In Galois theory , certain statements can be tightened if one assumes the linear disjointness of the intermediate bodies.
For example, which is Galois G ( MN / K ) of the compound word MN linearly disjoint intermediate body M , N are isomorphic to the product of the Galois G ( M / K ), G ( N / K ) of M and N . If we omit the linear disjoint, we only get the isomorphism of G ( MN / K ) to a subgroup of the product G ( M / K ) × G ( N / K ).
Related terms
- A field extension is regular if and only if is linearly disjoint to an algebraic closure of .
- An extension of a field of the characteristic is separable if and only if linearly disjoint to
- is.
literature
- Serge Lang , algebra . Springer-Verlag, New York 2002. ISBN 0-387-95385-X : Section VIII, §3
- Hideyuki Matsumura, Commutative ring theory . Cambridge University Press, Cambridge 1989. ISBN 0-521-36764-6 : Section 26