Well.. It looks like a obvious for me, But I want to check the below things right or not.
Let $K = SF(f/K)$ for separable polynomial $f(x) \in F[x]$ with its degree, $deg f = n(\geq2)$
(Here the $SF(f/F)$ means splitting field of $f(x)$ over the field $F$)
Say the set $A$ = $\{\alpha_1 \alpha_2,... \alpha_n\}$ is a set of the all roots of the $f(x)$
So my questions begin.
First Question)
For $\forall \sigma \in G(K/F) $, Is the bijective $\sigma : A \to A$ ?
(I.E. Are all the Galois group's elements bijective mapping between all roots of the $f(x)$?)
Second Qeustion)
Let's consider the case the $f(x)$ is not irreducible for a root $\alpha_i \in F$
I would say the $\{\alpha_1, \alpha_2 \}$ $\in F$ and $\{\alpha_3, ..., \alpha_n \} \in K-F$
Then, Is this true?
$\forall \sigma \in G(K/F) $, $\begin{cases} \sigma : \{\alpha_1, \alpha_2 \} \to \{\alpha_1, \alpha_2 \} & \text{} \\ \sigma : \{\alpha_3, ..., \alpha_n \} \to \{\alpha_3, ..., \alpha_n \} & \end{cases}$
(I.E. Should be roots which is elements of the each field "$F$" and "$K\setminus F$" mapped to $F$ and "$K\setminus F$" respectively by the bijective $\sigma : A \to A$?)
p.s.) Related with the second question Does it exist $\sigma(\in G(K/F)) : \alpha_1 \to \alpha_2$?
Thanks.