The theory of primitive polynomials over Galois rings is analogue to the same one over finite fields. It also provides useful tools for one to study the maximal period sequences over Galois rings. In the case of F<...The theory of primitive polynomials over Galois rings is analogue to the same one over finite fields. It also provides useful tools for one to study the maximal period sequences over Galois rings. In the case of F<sub>q</sub>, we have more complete results. In the case of Z<sub>p<sup>n</sup></sub>, n≥2, there are also some results. In particular, according to refs. [3, 4] and using the technique of trace representation of maximal period sequences over F<sub>q</sub>, we have found a discriminant which can judge whether a given polynomial f（x） over Z<sub>p<sup>n</sup></sub> is a primitive polynomial if f（x） mod p is a primitive polynomial over F<sub>p</sub>. Furthermore, it is easy to calculate the discriminant using the coefficients of f（x）.展开更多
文摘The theory of primitive polynomials over Galois rings is analogue to the same one over finite fields. It also provides useful tools for one to study the maximal period sequences over Galois rings. In the case of F<sub>q</sub>, we have more complete results. In the case of Z<sub>p<sup>n</sup></sub>, n≥2, there are also some results. In particular, according to refs. [3, 4] and using the technique of trace representation of maximal period sequences over F<sub>q</sub>, we have found a discriminant which can judge whether a given polynomial f（x） over Z<sub>p<sup>n</sup></sub> is a primitive polynomial if f（x） mod p is a primitive polynomial over F<sub>p</sub>. Furthermore, it is easy to calculate the discriminant using the coefficients of f（x）.
