# 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.