Polynomials and primitive elements in Galois rings

Abstract

A primitive element of the Galois ring GR( p n , m) is an element of the group U of units whose order is the exponent of U. We show that, for almost all odd prime powers p m where c divides p m − 1, if f is a polynomial over GR( p n , m) satisfying certain conditions, then there is a primitive element γ in the ring such that f(γ) is a cth power. We also show that for almost all Galois rings, if f satisfies certain conditions then f(α) is a primitive element for some α in the ring. These theorems are generalizations of Madden's results on primitive elements in finite fields.