A new method, which enables us to compute rather efficiently the Galois group of a polynomial over ℚ or ℤ, is presented. Reductions of this polynomial with respect to different prime modules are studied, and the information obtained is used for the calculation of the Galois group of the initial polynomial. This method uses an original modification of the Chebotarev density theorem, and it is in essence a probabilistic method. The irreducibility of the polynomial under consideration is not assumed. The appendix to this paper contains tables, which enable one to find the Galois group of polynomials of degree less than or equal to 10 as a subgroup of the symmetric group. This is the first part of the paper. The second part (the tables included) will be published in a subsequent issue. Bibliography: 9 titles.