I am trying to see how can I extend the automorphisms of the Galois extension $\mathbb{Q}(\sqrt{d})/\mathbb{Q}$, for $d$ square-free, to automorphisms of $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ that fix $\mathbb{Q}(\sqrt{d})$, when $n$ is given so that $\mathbb{Q}(\sqrt{d})\subseteq \mathbb{Q}(\zeta_n)$ .
I've found a result that tells me how $n$ has to be: if $d\equiv 1 \mod 4$ then $n$ can be taken as a multiple of $|d|$ and in other case as a multiple of $4|d|$ (where $|\cdot|$ is the absolute value).
Right now my progress is the following: I know that $\operatorname{Gal}(\mathbb{Q}(\sqrt{d})/\mathbb{Q})$ only contains the trivial automorphism and the one that sends $\sqrt{d}$ to $-\sqrt{d}$, and I have read that if $d<0$ then these automorphisms can only be extended to the trivial one and the one that acts as complex conjugacy respectively (but I don't understand this) and I want to know why and what happens if $d>0$.