Algebraic applications of Hurwitz' theorem

Notice that $\deg(R)=0$ and $g(Y)=1$ imply $g(X)=1$.

This means that every unramified cover of an curve of genus $1$ is again a curve of genus $1$. Translating into the algebraic language and using the fact that , if $\textrm{char}(k) \neq 2,3$, any curve of genus $1$ has a birational model of the form $y^2=x^3+px+q$, with $4p^3+27q^2 \neq 0$, we obtain the following result:

Assume $\textrm{char}(k)\neq 2,3$ and let $p, q \in k$ with $4p^3+27q^2 \neq 0$. Then every finite, unramified extension of the quotient field of

$k[x,y]/(y^2-x^3-px-q)$

can be written as the quotient field of

$k[s,t]/(s^2-t^3-at-b)$,

for some $a,b \in k$ with $4a^3+27b^2 \neq 0$.

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 $F|K$.

On this other thread, about elliptic curves about function fields, the functional analogue of Mazur's theorem was evoked, and described as a simple consequence that the genus of modular curves grows with the conductor. This is indeed an application of the inequality $g(X)\geq g(Y)$ you mention.