Consider the following theorem$^{*}$:
Theorem of conservation of number: Suppose $f:X \longrightarrow Y$ is an étale
covering, and $Y$ is connected. Then the number of points in a fibre $f^{-1}(y)$ is
independent of $y \in Y$.
The way of proof is that locally on $Y$, the variety $X$ can be defined in $\mathbb{A}^1 \times Y$ by one equation, say $T^m + a_1T^{m-1} + ... + a_m = 0$, where the $a_i \in K[Y]$. Now, since $f$ is étale, all the roots of the equation $T^m + a_1T^{m-1} + ... + a_m = 0$ are simple, and there are precisely $m$ of them.
My question is why all the roots of the equation $T^m + a_1T^{m-1} + ... + a_m = 0$ are simple, and there are precisely $m$ of them $\Longrightarrow$ the number of points in a fibre $f^{-1}(y)$ is independent of $y \in Y$? How to connect the simple $m$ roots of the equation with the number of points in a fiber?
$^*$Algebraic Geometry I - Algebraic Curves - Algebraic Manifolds and Schemes (I. R. Shafarevich). Section 5.5.
thanks