Let $K/F$ be Galois with Galois group $G$ and let $a\in K$. Suppose $\sigma_1,...,\sigma_n$ are all the elements of $G$ and that $\sigma_1(a),...,\sigma_n(a)$ are all distinct. Then do we have that $K=F(a)$? Is this true because $G$ would then be uniquely determined by where it maps $a$? Is there a more explicit way to phrase this?
Further, if we were to assume all the images of $a$ under $G$ were distinct, why would this prevent us from having $K=F(a,b)$ for some other element $b\in K$?