Proposition 7

— Let E and F be two extensions of K contained in $\Omega$; if E and F are linearly disjoint over K, then every subextension of E and every subextension of F are linearly disjoint over K. Conversely, if for every pair of finitely generated subextensions E', F' of E and F respectively, E' and F' are linearly disjoint over K, then E and F are linearly disjoint over K.

For the condition for E and F to be linearly disjoint over K may be expressed thus: if $(a_\alpha)$ is a free family in E and $(b_\beta)$ a free family in F, then the relation $\sum_{\alpha,\beta} \lambda_{\alpha\beta} a_\alpha b_\beta = 0$, where $\lambda_{\alpha\beta} \in K$, implies $\lambda_{\alpha\beta} = 0$ for each pair of suffixes. But this condition is satisfied for each pair of free families if it holds for each pair of finite free families.

Thus we may say, speaking intuitively, that linear disjunction is a property « of finite character ».