Tame distillation and desingularization by p-alterations

We strengthen Gabber's l'-alteration theorem by avoiding all primes in- vertible on a scheme. In particular, we prove that any scheme X of finite type over a quasi-excellent threefold can be desingularized by a char(X)- alteration, i.e., an alteration whose order is only divisible by primes nonin- vertible 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 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 k of residue characteristic p has no nontrivial p-extensions, then any algebraic extension l/k is tame.