Minimal polynomial of intermediate extensions under Galois extensions.

Solution 1:

1. There is no guarantee that $z_i(a)\in F$ for all $a$; in fact, it will never be the case unless $a\in F$. But you don't need each to be in $F$, you need the coefficients of the product to be in $F$.

   In order to show that the coefficients of the product are in $F$, it suffices to show that for every $\sigma\in G$ we have $$\sigma\left(\prod_{i=1}^r(x-z_i(a))\right) = \prod_{i=1}^r\left(x - \sigma(z_i(a))\right) = \prod_{i=1}^r(x-z_i(a)),$$ because the coefficients lie in $F$ if and only if the coefficients are invariant under the action of the Galois group.

2. If, say $z_iH = \mathrm{id}_GH$, then that means that $z_i\in H$. But $H$ is precisely the subgroup of elements that fix $F(a)$. So what is $z_i(a)$ then?