Proposition 4

— Let $(N_i)_{i \in I}$ be a family of quasi-Galois extensions of $K$ contained in $\Omega$. Let $N = \bigcap_{i \in I} N_i$ and $M = K\left( \bigcup_{i \in I} N_i \right)$; then the extensions $N$ and $M$ of $K$ are quasi-Galois.

Let $u$ be a $K$-automorphism of $\Omega$. We have $u(N_i) = N_i$ for all $i \in I$, hence we obtain the equalities $u(N) = N$ and $u(M) = M$; now the proposition follows from Cor. 1 (V, p. 54).

Let $A$ be a set of elements of $\Omega$, and let $B$ be the set of elements of $\Omega$ which are conjugates of elements of $A$; expressed differently, if $G$ is the group of $K$-automorphisms of $\Omega$, we have $B = \bigcup_{u \in G} u(A)$. For each $u \in G$ we have $u(B) = B$ whence $u(K(B)) = K(B)$. Therefore $K(B)$ is a quasi-Galois extension of $K$ containing $A$ and it is immediate that every quasi-Galois extension $N$ of $K$ containing $A$ contains $B$, hence $K(B)$. We shall say that $K(B)$ is the quasi-Galois extension generated by $A$. This definition applies in particular when $A$ is an extension of $K$.

The next proposition follows directly from the preceding remarks.