Remark 2

Suppose that $E$ is a quasi-Galois extension of $K$ and $F$ a quasi-Galois extension of $E$; it is not always the case that $F$ is quasi-Galois over $K$ (V, p. 153, Ex. 1). The reason is the following: let $u$ be a $K$-automorphism of $\Omega$; we have $u(E) = E$, but if $f$ is the minimal polynomial over $E$ of an element $x$ of $F$, it is not necessarily invariant under $u$; therefore $u(x)$ is not necessarily conjugate to $x$ over $E$, and so need not necessarily belong to $F$. In that case $F$ and $u(F)$ are two distinct quasi-Galois extensions of $E$ which are $K$-isomorphic but not $E$-isomorphic.