How to compute the normalization of $R:=\frac{k[X,Y]}{(Y^2-f(X))}$ with $f(X)\in k[X]$ of odd order?
I proved that $R$ is normal iff $f(X)$ is square free. What are the methods and the ideas that bring me to the normalization of a ring?
How can i describe the map $\Omega_{R/k}\rightarrow \Omega_{\bar{R}/k}$?
Thank you :)
In general if $f(x)=p(x)^2\cdot h(x)$ where $h(x)$ is the square-free part of $f(x)$ we have $\frac{y^2}{p(x)^2}=h(x)$ in the fraction field of $R$ hence you have to adjoin the inverse of $p(x)$ to the ring to get the normalization.
Hence the normalization is $k[x,y,z]/(y^2-f(x),z\cdot p(x)^2-1)$