  We strengthen Gabber's $l'$-alteration theorem by avoiding all primes
invertible on a scheme. In particular, we prove that any scheme $X$ of finite
type over a quasi-excellent threefold can be desingularized by a
$\mathrm{char}(X)$-alteration, i.e. an alteration whose order is only divisible
by primes non-invertible on $X$. The main new ingredient in the proof is a tame
distillation theorem asserting that, after enlarging, any alteration of $X$ can
be split into a composition of a tame Galois alteration and a
$\mathrm{char}(X)$-alteration. The proof of the distillation theorem is based
on the following tameness theorem that we deduce from a theorem of M. Pank: if
a valued field of residue characteristic $p$ has no non-trivial $p$-extensions
then any its algebraic extension is tame.
