Resolution of singularities

**Resolution of singularities fails over imperfect fields.** If the field $k$ is not perfect, then there is an integral scheme $X$ of dimension zero that is separated and of finite type over $k$ and has no resolution of singularities: no integral scheme smooth over $k$ admits a proper birational morphism to $X$.

A witness: $k$ has characteristic $p > 0$ and there is some $a \in k \setminus k^p$. Take $X = \operatorname{Spec} k(a^{1/p})$. Every integral scheme birational to $X$ has function field $k(a^{1/p})$, which is not separable over $k$, whereas the function field of an integral scheme smooth over $k$ is separable over $k$.

So the hypothesis PerfectField k cannot be dropped from resolution_of_singularities or from resolution_of_singularities_of_topologicalKrullDim_le_three.