Let $\pi:Y\to X$ be a Galois cover, i.e. a finite morphism of nonsingular varieties over an algebraically closed field $\Bbbk$ such that $K(X)\hookrightarrow K(Y)$ is Galois. Let $H\subset X$ be the branch locus of $\pi$. We assume that each component of $H$ is nonsingular^{1}. I seek your help in proving the following statement:

PropositionLet $P\in X$ be a regular point of $H$ and let $\eta$ be the generic point of the component $Z$ of $H$ which contains $P$. Assume that every point $\tau$ of codimension one with $\pi(\tau)=\eta$ has ramification index $n$. Then, $\left|\pi^{-1}(P)\right|=\deg(\pi)/n$.

Of course, if you can refer me to a text where this or a more general result is proven, that'd be great, since I couldn't find anything. Also, feel free to provide a proof which is different from my approach, but this is what I did so far:

- Restrict to the case where $X=\mathrm{Spec}(A)$ and $Y=\mathrm{Spec}(B)$ are affine (duh).
- Use the primitive element theorem and further localization to get $B=A[t]$ for some $t$ which is integral over $A$ and the minimal polynomial $F\in A[x]$ of $t$ has degree equal to $\deg(\pi)$.
- Denote by $F_P$ the image of $F$ under $A[x]\twoheadrightarrow (A/\mathfrak{m}_P)[x]=\Bbbk[x]$, then for any $Q\in\pi^{-1}(P)$, we have $0=F(t)(Q)=F_P(t(Q))$. The cardinality of the fiber of $P$ is therefore equal to the number of
*distinct*zeros of $F_P$. - I thought I could write $F=\prod_i (x-f_i)$ for certain $f_i\in B$ by possibly localizing further, since $K(Y)$ is normal over $K(X)$. These $f_i$ will all be distinct, but at $P$, exactly $n$ of them should have the same value. I don't know how to prove that, though.
- I also thought about considering the discriminant of the $(k-1)$-st formal derivative of $F$ - this is a function on $Y$ which 'knows' where $F$ has $k$-uple zeros. I have even less of an idea how to relate that to ramification, though.

^{1} Of course, if you want to weaken these assumptions anywhere, be my guest.

necessarilythe same ramification index. Of course, this is false for non-Galois covers. $\endgroup$ – Francesco Polizzi Aug 5 '11 at 8:31genericstabilizer along $Z$ has order $n$. Why can't the order jump at $P$? $\endgroup$ – Laurent Moret-Bailly Aug 7 '11 at 10:39