Hurwitz' theorem states that for a finite separable morphism $f : X \to Y$ of curves of degree $n$ and with ramification divisor $R$, we have


$2 g(X) - 2 = n (2 g(Y) - 2) + \deg(R)$.


Besides, we have $\deg(R)=\sum\_{p \in X} (e\_p - 1)$ if $f$ has only tame ramification [Hartshorne, IV, Cor. 2.4]. One of the consequences is $g(X) \geq g(Y)$, which also holds when $f$ is not supposed to be separable [loc. cit. 2.5.4].


One purely algebraic application of Hurwitz' theorem is Luroth's theorem, which states every nontrivial intermediate field of $k(t)$ over $k$ is isomorphic to $k(x)$ over $k$. However, it is easy to give a direct algebraic proof of Luroth's theorem, even if $k$ is not supposed to be algebraically closed (which is probably needed for Hurwitz' theorem). Therefore I wonder if there are other algebraic application of Hurwitz' theorem using the correspondence between curves and function fields.


**Question:** Are there other interesting algebraic applications of Hurwitz' theorem?



Algebraic applications of Hurwitz' theorem
As far as I remember the Riemann-Hurwitz-formula is used to prove the inequality


$|\mathrm{Aut}(F|K)|\leq 84(g-1)$


for the number of automorphisms of an algebraic function field $F$ of one variable over $K$,
where $K$ has characteristic $0$ and $g\geq 2$ holds for the genus of $F|K$.


