I would like to know if the lemma below is correct. In such a case, a reference would be also useful. Thanks
Lemma. Let $L / K$ be a finite Galois extension and let $g \in G(L / K)$ (the Galois group of $L / K$). Let $f \in L[x]$ be an irreducible separable polynomial of degree $n > 0$ and let $a$ be a root of $f$ and $b$ be a root of $gf$ (in some algebraic closure of $L$). Then we can extend $g$ to $\tilde{g} \in G(L(a) / K)$ such that $\tilde{g}(a) = b$.
Proof (sketch): The fundamental theorem of Galois theory asserts (among other things) that $G(L(a) / K) / H \cong G(L / K)$ via the map $h H \to h|_L$, where $H = G(L(a) / L)$. So asking that $\tilde{g}|_L = g$ amount to asking that $\tilde{g}$ belongs to a fixed coset of $G(L(a) / K) / H \cong G(L / K)$. Each coset has $|H| = [L(a) : L] = n$ (since $f$ is irreducible) element, hence at least one $\tilde{g}$ satisfies $\tilde{g}(a) = b$. $\square$