Irreducible Polynomials Which Divide Trinomials Over GF(2). - The ...

This computationally useful criterion, due to L.R. Welch, determines whether theirreducible polynomials of primitivity t divide trinomials, for any odd integer t > 3,without directly identifying which irreducible polynomial divides which trinomial.2.2 Further Theorems and ResultsWhen the primitivity t is a prime p, letΦ p (x) = (xp − 1)(x − 1) = f 1(x)f 2 (x) . . . f r (x)be the factorization of the p th cyclotomic polynomial into irreducible factors over GF(2).Here the indexr = φ(p)/n = (p − 1)/nis the number of irreducible factors of Φ p (x) over GF(2), and (p − 1)/r is the order of2 in the multiplicative group modulo p. Again, all the f i (x)’s have the same degree(say, n) and the same primitivity p, and they are all the irreducible polynomials havingprimitivity p over GF(2). Let α be a root of f i (x). The following results relate to theindex r.Definition 2. Let f(x) be an irreducible polynomial of degree n. The reciprocal of f(x)is defined asf ∗ (x) = x n f( 1 x ). 16