Tensor product of fields

In abstract algebra two intermediate body and a body extension called linearly disjoint if each set of items of which is independently linear, also linearly independent. An equivalent characterization is: The Figure

Is injective ( for notation see tensor product ). In this description, one can see immediately that linear disjointness is a symmetric property of and.

The average linearly disjoint extensions is always part of the body, that is,

The converse is not true in general, but at least when one of the two extensions and is finite and Galois.

In the Galois theory can exacerbate certain statements, if we assume the linear disjointness of the intermediate body.

For example, the Galois group G ( MN / K) is the compound word of MN linearly disjoint intermediate body M, N is isomorphic to the product of the Galois groups G (M / K ), G (N / K ) of M and N. Leaving the linear disjointness away, you 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 it is linearly disjoint from an algebraic degree of.
• An extension of a field of characteristic separable if and only if linearly disjoint from