Proposition 8

— Let E, F, G be three extensions of a field K contained in $\Omega$, such that $F \subset G$. For E and G to be linearly disjoint over K it is necessary and sufficient that E and F should be linearly disjoint over K and that E(F) and G be linearly disjoint over F.

Fig. 2.

The condition is necessary: suppose that E and G are linearly disjoint over K. The same then holds of E and F (Prop. 7); on the other hand, if B is a basis of E over K, this is also a basis of the algebra F[E] over F; since B is by hypothesis free over G, F[E] and G are linearly disjoint over F, and the same is true of $E(F) = F(E)$ and G, by Prop. 6.

The condition is sufficient: with the same notation it implies that B is free over F, hence it is a basis of F[E] over F; since F[E] and G are by hypothesis linearly disjoint over F, B is free over G, and this shows that E and G are linearly disjoint over K.