Proposition 3

— The restriction homomorphism of $\mathrm{Gal}(N/K)$ into $\mathrm{Gal}(L/K)$ is surjective.

More generally, consider two subextensions L and L' of N, and a K-isomorphism $u$ of L onto L'. Choose an algebraic closure $\Omega$ of K containing N as subextension (V, p. 23, Th. 2). There exists a K-automorphism $v$ of $\Omega$ which coincides with $u$ on L (V, p. 52, Prop. 1), and since N is a quasi-Galois extension of K, $v$ induces a K-automorphism $\sigma$ of N (V, p. 55, Remark 1). In other words, the element $\sigma$ of $\mathrm{Gal}(N/K)$ coincides with $u$ on L.