I am trying to solve Exercise IV.2.3 from Liu's book. I want to prove that $K[x,y]/(x^2+y^3+t^n)$ is regular, where $K$ is the fraction field of a discrete valuation ring and $t$ is a uniformizing parameter.
I would like to apply the Jacobian criterion, but I do not know if $K$ is perfect.
Finding explicitly all maximal ideals and study the localizations seems too long, but maybe it is the way to go?
Any help would be appreciated.