Number of elements in fiber My Question: If we have $f:X\to Y$ an etale morphism and we assume
$X,Y$ smooth affine Varieties,
why is it true, that $|f^{-1}(y)|\leq deg(f)$ ?
Why isn´t there any point of $Y$, which has more elements in the fiber than the degree of $f$?
 A: If the  morphism of varieties $f:X\to Y$ is étale it is automatically quasi-finite.
One of the versions of Zariski's main theorem (see our friend Akhil's notes, Theorem 8.5) then implies that $f$ can be factored as $f=f'\circ j: X \stackrel {j} {\hookrightarrow}    Y' \stackrel {g} {\rightarrow} Y$ where $j$ is an open immersion and $g$ is finite.
Since you know the desired relation for the finite morphism $g$ and since $j$ is of course injective you get $$|f^{-1}(y)|\leq |g^{-1}(y)|\leq deg(g)=deg(f)$$
just as you wished. 
Remark
It is not necessary to assume smoothness of the varieties and only quasi-finiteness [Qing Liu, Chapter 4, Proposition 3.23 (a)] is used from the étale hypothesis.
A: I give an hint, in the form of the following question : let $A$ and $B$ be $k$-algebras of finite type over $k$ that are domains, such that $B$ is a finite $A$-algebra of degree $d$ that is formally étale, and let $\mathfrak{p}$ be a prime ideal of $A$, how many primes ideals $\mathfrak{q}$ of $B$ are lying over $\mathfrak{p}$ ? 
