Computation of the Galois group of a polynomial with rational coefficients

A new method, which enables us to compute rather efficiently the Galois group of a polynomial over $\mathbb{Q}$, respectively, over $\mathbb{Z}$ is presented. Reductions of this polynomial with respect 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 probability 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.
Here the first part of the paper is published. The second part (the tables included) will be published in the next issue.