Galois extension - minimum polynomial

Question

Let $K$ be a Galois extension of $F$ and let $a\in K$. Let $n=[K:F]$, $r=[F(a):F]$, $G=\text{Gal}(K/F)$ and $H=\text{Gal}(K/F(a))$.

We symbolize with $\tau_1, \ldots , \tau_r$ the representatives of the left cosets of $H$ in $G$.

1. Show that $\displaystyle{\min (F,a)=\prod_{i=1}^r\left (x-\tau_i(a)\right )}$.

2. Show that $\displaystyle{\prod_{\sigma \in G}\left (x- \tau_i(a)\right )=\min (F,a)^{n/r}}$, where $\tau_i$ is the unique representative such that $\sigma \in \left[ \tau_i \right]$.

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Could you give me a hint how we could show these two points? I don't really have an idea.

Answer 1

For the first point, try to show that if $i \neq j$, then $\tau_i(a) \neq \tau_j(a)$.

For the second one, try to show that if $\sigma, \tau$ belong to the same class modulo $H$, then $\sigma(a) = \tau(a)$