Given the twentieth root of unity a splitting for it's minimal polynomial is clearly $\Bbb Q(w_{20})$ where $w_{20}$ is the twentieth primitive root of unity. As the isomorphisms only map to roots of the minimal polynomial it only maps to primitive roots and so the Galois group is $G\cong \Bbb U_{20}$ I have a question about one of the Galois correspondences between the subfields of $\Bbb Q(w_{20})$ and the subgroups of $G$ in particular though . Consider the subgroup $\langle 9 \rangle$.
It seems to me that the element $w+w^9$ would be fixed by this group as (denoting $\sigma$ the automorphism which generates this group ) $\sigma(w+w^9)=w^9+w^{81}=w^9+w^1$. However in my course notes it says that the element fixed by this subgroup is $w^2+w^{18}$. I can see from applying the automorphism that this is indeed fixed but I don't understand why my choice of element wasn't valid .. Could anyone please explain this to me ?