Field compositum

Inside a common overfield, the compositum $L_1{L}_2$ is the smallest field containing both $L_1$ and $L_2$.

The multiplication image of the tensor product of fields in a common overfield generates the compositum as a field. For algebraic extensions that image is already a field; in general it need not be. The compositum is therefore distinct from the tensor product algebra.

More generally, two field extensions inside a common overfield are linearly disjoint field extensions when the multiplication map from their tensor product of fields into that overfield is injective. The following basis criterion is the finite-degree case.

Finite extensions $L_1/K$ and $L_2/K$ are linearly disjoint when products of basis elements form a basis of their compositum. In particular,Extensions of coprime finite degrees are linearly disjoint.