We say that embedded resolution of singularities in dimension $m$ holds over a field $k$ if for every quasi-projective regular variety $Y$ over $k$ of dimension $m$ and every proper closed subset $Z$ of $Y$, there is a projective morphism $\pi:Y^{\prime}\to Y$ of quasi-projective regular varieties over $k$ such that the set $\pi^{-1}(Z)$ is the support of a normal crossing divisor and such that $\pi$ is an isomorphism over $Y\setminus Z$.