Factorization of polynomials over finite fields

Abstract

The laws of factorization of irreducible polynomials with integer coefficients over finite fields, a long-standing problem of number theory and algebra. The various reciprocity laws of number theory are connected with this problem. The Galois group of an irreducible polynomial $f(x)$ of degree n over the field of rational numbers, consider as a subgroup of the symmetric group $S_{n}$, actually describes possible types of factorization of $f(x)$ with respect to simple modules. The next problem is to describe prime numbers giving a certain type of factorization of the polynomial $f(x)$ in terms of invariants associated with this polynomial. For polynomials with Abelian Galois group this problem is solved in principle by a dap class field theory. For polynomials with a non-Abelian Galois group, little is known for certain classes of polynomials. In this paper we propose a method for solving this problem for irreducible over the field rational numbers of cubic polynomials.