Proposition 1

— Let N be an extension of K and $(N_i)_{i \in I}$, a non-empty family of subextensions of N. We put $E = \cap_{i \in I} N_i$ and $F = K \left( \bigcup_{i \in I} N_i \right)$. If all the extensions N, are Galois over K then the same is true of E and F.

In the first place E is algebraic and separable over K (V, p. 36, Prop. 1) and the same holds for F (V, p. 41, Prop. 8). Moreover, E and F are quasi-Galois over K by Prop. 4 (V, p. 55).