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 ${\ displaystyle M}$ ${\ displaystyle N}$ ${\ displaystyle L / K}$ ${\ displaystyle M}$ ${\ displaystyle K}$ ${\ displaystyle N}$

{\ displaystyle M \ otimes _ {K} N \ to L}

is injective (see tensor product for notation ). This description also shows immediately that linear disjointness is a symmetric property of and . ${\ displaystyle M}$ ${\ displaystyle N}$

The intersection of linearly disjoint partial extensions is always the basic body , i.e. H. ${\ displaystyle K}$

{\ displaystyle M \ cap N = K.}

The converse is not generally true, but at least if one of the two extensions and is finite and Galois . ${\ displaystyle M / K}$ ${\ displaystyle N / K}$

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 . ${\ displaystyle L / K}$ ${\ displaystyle L}$ ${\ displaystyle {\ bar {K}}}$ ${\ displaystyle K}$

An extension of a field of the characteristic is separable if and only if linearly disjoint to ${\ displaystyle L}$ ${\ displaystyle K}$ ${\ displaystyle p> 0}$ ${\ displaystyle L}$

{\ displaystyle K ^ {p ^ {- \ infty}} = \ {x \ in {\ bar {K}} \ mid \ exists n \ colon x ^ {p ^ {n}} \ in K \}}

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