question on Galois theory Can anybody help me with the following question ? 
I start with a number field $F/\mathbb{Q}$ which is abelian (that is, a Galois extension of abelian Galois group). I know by the Kronecker-Weber theorem that $F$ can be embedded in some cyclotomic field $\mathbb{Q}(\mu_d)$. Pick an element $\sigma \in \mathrm{Gal}(\mathbb{Q}(\mu_d)/ \mathbb{Q})$.
My question is: why is it true that $\sigma(F) \subset F$? 
Thanks for your help. 

Rough answer: By the Galois correspondence subgroups of the (cyclotomic) Galois group correspond to intermediate fields. But also normal subgroups correspond to normal extensions.
Since everything in sight is an abelian, all subgroups of the cyclotomic Galois group are normal, hence all intermediate fields form normal extensions of $\mathbb{Q}$.
Now restrictions of automorphisms to intermediate fields make monomorphisms into $\mathbb{C}$. But normality means this monomorphism is actually into $F$ (since take a generator for $F/\mathbb{Q}$, then monomorphisms act on its conjugates, which all lie in $F$ by normality).
