Assume E and Fare the quotient fields of integral domains Rand S respectively. Then C is an R-module and an S-module in the obvious way 

(a) E and F are linearly disjoint over K if and only if every subset of R that is linearly independent over K is also linearly independent over S.

(b) Assume further that R is a vector space over K. Then E and Fare linearly disjoint over K if and only if every basis of R over K is linearly independent over F.

(c) Assume that both R and S are vector spaces over K. Then E and Fare linearly disjoint over K if and only if for every basis X of R over K and basis Y of S over K, the set {uv I u ϵ X; v ϵ Y} is linearly independent over K.

E and Fare always extension fields of a field K, and C is an algebraically closed field containing E and F.