The author gives an algorithm for factorizing a multivariate polynomial \(f\) over a local field \(K\) in time polynomial in the length of the input data and the characteristic of the field. He reduces the problem to the case of a separable polynomial in one variable, for which the theorem was already proven by himself [J. Sov. Math. 34, 1838--1882 (1986); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 137, 124--188 (1984; Zbl 0561.12010)]. He uses the Newton polygon to embed \(K[X]/(f)\) into a product \(\prod_{\ell \in L}K_{\ell}[X]/(f_{\ell}),\) where \(K_{\ell}/K\) is tamely ramified and \(f_{\ell}\) is irreducible over \(K_{\ell}\). Finally he uses the Newton polygon to compute the roots of \(f\) in \(K_{\ell}[X]/(f_{\ell})\). Finally he computes the irreducible factors by taking norms.