A criterion for primitive polynomials over Galois rings (Q5917762)

A finite commutative local ring \((R,M)\) with identity is called a Galois ring if it is an unramified extension of a prime ring \(\mathbb{Z}_{p^n}\). Let \(k = R/M = F_{p^r}\) be the residue class field of \(R\), \(\mu : R[x] \to k[x]\) the induced homomorphism of the polynomial rings by the natural homomorphism of \(R\) into \(k\). An irreducible polynomial \(f \in R[x]\) is called basic irreducible if \(\mu f\) is irreducible in \(k[x]\). Let \(f(x) \in R[x]\) be a monic basic irreducible polynomial with degree \(m \geq 2\). The period of \(f\) divides \((q^m - 1) p^{n - 1}\) where \(q = p^r\). Call \(f\) a primitive (subprimitive) polynomial of degree \(m\) if \(\text{per} (f) = (q^m - 1) p^{n - 1} ((q^m - 1) p^{n - 2}\) respectively). A criterion for \(f\) to be primitive (subprimitive) is found in this paper.