Proposition 6

Let E and F be two extensions of K contained in $\Omega$, and A and B two subalgebras of $\Omega$ over K such that E is the field of fractions of A and F the field of fractions of B. Then for E and F to be linearly disjoint over K it is necessary and sufficient that A and B be linearly disjoint over K.

The condition is clearly necessary. Conversely, if A and B are linearly disjoint over K, A and F are so too, because if a family of elements of $\Omega$ is free with respect to B, it is free with respect to the field of fractions F of B (II, p. 315, Cor. 1 and p. 316, Cor. 3); now the same argument shows that E and F are linearly disjoint over K.

$^1$ It suffices to consider for example the case where $\Omega$ is the field $K(X, Y)$ of rational fractions in two indeterminates X and Y and $E = K(X), F = K(Y)$.