A criterion for primitive polynomials over Galois rings

In this paper, we present a link between the representation of a root of a basic irreducible polynomial f(x) over a Galois ring and its order, and derive algebraic discriminants for primitive polynomials and sub-primitive polynomials, respectively. The principal parts of these discriminants are determined by the coefficients of f(x)modp and f(x)modp2, respectively. By these results, we can give some fine criteria for primitive polynomials over Galois rings with characteristic 2n, and characterize trinomial and pentanomial primitive polynomials over Z2n completely.