Exercise 6, § 14

Let $\Omega$ be an algebraically closed extension of a field K, and N a subextension of $\Omega$. Show that if N is quasi-Galois over K and if E and E' are two conjugate extensions of K contained in $\Omega$, then N(E) and N(E') are conjugate over K. Conversely, if N has this property and if tr . $\deg_K N$ is finite and < tr . $\deg_K \Omega$, then it is necessarily algebraic and quasi-Galois over K.