I have understood completely the reasoning above Proposition 2.7. But I cannot understand how to prove the proposition itself. Let me ask my questions more precisely?
1) In the reasoning above $\beta$ is a root of $p^{\sigma}$ not a root of $p$. And this confuses me.
2) Why the number of possible extensions is $\leq$ number of roots of $p$? Why it cannot be bigger?
3) How to show that distinct roots give different extensions?
Unfortunately, I cannot to prove this statment in rigorous way and I would be very grateful if anyone can show its proof. I would like to see detailed proof.
In the reasoning $\beta$ is a root of $p^{\sigma}$. Nothing at all is said about whether it is a root of $p$ or not; that is immaterial. (It could be a root of $p^{\sigma}$ as well, e.g., if $\sigma$ is the identity).
$p^{\sigma}$ has the same number of roots (and of distinct roots) as $p$ in $L$. Since the image of $\alpha$ must be a root of $p^{\sigma}$, and completely determines the extension, the number of potential images of $\alpha$ is at most the number of distinct roots of $p^{\sigma}$, which is the same as the number of distinct roots of $p$.
Because if $\beta\neq\beta’$, then the image of $\alpha$ under one extension is $\beta$ and under the other extension is $\beta’$. Given that the two extensions send $\alpha$ to different things, the two extensions are different.