Elementary proof of the Hurwitz formula I am aware of two forms of the Hurwitz formula. The first is more common, and deals only with the degrees. So if $f:X \rightarrow Y$ is a non-constant map of degree $n$ between two projective non-singular curves, with genera $g_X$ and $g_Y$, then 
$$
2(g_X-1) = 2n(g_Y-1) + \deg(R),
$$
where $R$ is the ramification divisor of $f$.
The proof of this was given to me as an exercise when I started my PhD, and I am very happy with it.
However, in some other work that I was doing it appeared that one could strengthen this to say rather that if $K_X={\rm div}(f^*(dx))$ and $K_Y={\rm div}(dx)$ are canonical divisors of $X$ and $Y$, then 
$$
K_X = n\cdot K_Y + R.
$$
I have found this alluded to in a number of places, and even stated in Algebraic Curves Over Finite Fields by Carlos Moreno. However, this was without proof, and every idea of a proof that I have seen is in sheaf theoretic language. I am slowly getting through sheaves and schemes, but I am currently trying to prove this in an elementary manner (fiddling around with orders of $dx$ etc.), in the wildly ramified case (the tamely ramified case is fine).
I would like to know if


*

*It is possible to prove this without using sheaves etc in the wildly ramified case

*If so, are there any references that would help with this.


edit: Also, is there a different name for the "more specific" Hurwitz formula?
 A: You might like Griffiths's proof of Riemann-Hurwitz (Introduction to Algebraic Curves page 91, theorem (8.5) ) for a morphism $f:X\to Y$ .
It is very natural: he starts from a meromorphic differential form $\omega \in \Omega(Y)$ (whose  existence he provisionally admits  )  , lifts it to $f^*\omega \in \Omega(X)$, then  computes and compares the divisors $div(f^*\omega), f^*(div(\omega))$ and $R$ locally in coordinates.
Riemann-Hurwitz is not difficult but somewhat explains the slightly confusing  result that taking  divisors and lifting  do not commute for differential forms.
Edit
Stichtenoth  gives a proof for fields of any characteristic in Corollary 3.4.13 in the language of function fields, but only under a separability assumption.
The most general formula I'm aware of (but I'm not in the least a specialist) and  which does not require separability assumptions is Tate's Genus Formula.
You can find it as  Corollary 9.5.20 in Villa's Topics in the Theory of Algebraic Function Fields.
A: This formula appeared in Miranda's book Algebraic Curves and Riemann Surfaces, page 135. It does not require a proof by shreaves. What you need is just basic definition. Looking at that page and previous pages should give you enough directions to prove this yourself. 
Edit: The more "specific" formula you are looking for is just this formula for differential forms. 
