Periods of polynomials over a Galois ring

The period of a monic polynomial over an arbitrary Galois ring GR(pe, d) is theoretically determined by using its classical factorization and Galois extensions of rings. For a polynomial f(x) the modulo p remainder of which is a power of an irreducible polynomial over the residue field of the Galois ring, the period of f(x) is characterized by the periods of the irreducible polynomial and an associated polynomial of the form (x−1)m + pg(x). Further results on the periods of such associated polynomials are obtained, in particular, their periods are proved to achieve an upper bound value in most cases. As a consequence, the period of a monic polynomial over GR(pe, d) is equal to pe−1 times the period of its modulo p remainder polynomial with a probability close to 1, and an expression of this probability is given.