Factorization of irreducible polynomials over a finite field with the substitution for x
The paper presents a careful analysis of the Cantor-Zassenhaus polynomial factorization algorithm, thus obtaining tight bounds on the performances, and proposing useful improvements. In particular, a new simplified version of this algorithm is described, which entails a lower computational cost. The key point is to use linear test polynomials, which not only reduce the computational burden, but can also provide good estimates and deterministic bounds of the number of operations needed for factoring....
1. Introduction. Let K be a field of characteristic p ≥ 0 and let f(X) be a polynomial of degree at least two with coefficients in K. We set f₁(X) = f(X) and define for all r ≥ 1. Following R. W. K. Odoni [7], we say that f is stable over K if is irreducible over K for every r ≥ 1. In [6] the same author proved that the polynomial f(X) = X² - X + 1 is stable over ℚ. He wrote in [7] that the proof given there is quite difficult and it would be of interest to have an elementary proof. In the sequel...
In this note we generalize some results from finite fields to Galois rings which are finite extensions of the ring Zpm of integers modulo pm where p is a prime and m ≥ 1.
Let , where and , and let be a sequence of integers given by the linear recurrence for . We show that there are a prime number and integers such that no element of the sequence defined by the above linear recurrence is divisible by . Furthermore, for any nonnegative integer there is a prime number and integers such that every element of the sequence defined as above modulo belongs to the set .