Let $X$ be an affine variety. Let $A$ be the coordinate ring of $X$ and let $K$ be the fraction field of $A$. Given a Galois extension $K\subset L$, let $B$ be the integral closure of $A$ in $L$. Let $Y$ be the associated affine variety of $B$.

Questions: can rational (Gorenstein) singularities be inherited from $X$ to $Y$?

nothold for rational singularities though. Although in the Gorenstein case, rational singularities are equivalent to log terminal singularities... (and log terminal always implies rational). $\endgroup$ – Karl Schwede Jul 12 '14 at 3:46