Let $K$ be a field,

$f(x) \in K[x]$ be a monic polynomial with distinct roots, $\deg(f)=d$.

Let $R=K[x,y]/(y^n-f(x))$ and $C=Spec(R)$. $\:\:\;\:\:\:\:\quad$ ($n>2$ integer)

What is the basis of the first de Rham cohomology $H_{dR}^1(C)$?

I solved it for $n=2$ and wonder how it could be generalized. If it is hard, what if $n=3$ or $n$ is prime.

For case $n=2$: you can check Kiran S. Kedlaya- p-adic cohomology



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