### Some classes of irreducible polynomials

Anca Iuliana Bonciocat, Nicolae Ciprian Bonciocat
2006 Acta Arithmetica
1. Introduction. Lipka [10] obtained some irreducibility criteria for integer polynomials of the form f (X) = a n X n +a n−1 X n−1 +· · ·+a 1 X +a 0 p k with a 0 a n = 0, p a prime number and k a positive integer. For instance, he proved that for fixed p, a 0 , a 1 , . . . , a n with a 0 a 1 a n = 0, f is irreducible over Q for all but finitely many positive integers k. Another criterion proved in [10] is that given integers a 0 , a 1 , . . . , a n with a 0 a n = 0, the polynomial a n X n + a
more » ... omial a n X n + a n−1 X n−1 + · · · + a 1 X + a 0 p is irreducible over Q for all but finitely many prime numbers p, this result being a consequence of a theorem of Ore [14, Th. 5, p. 151]. These results can be formulated equivalently as in the earlier paper of Weisner [23] : If a polynomial f (X) ∈ Z[X] has a simple rational root, then for a fixed integer c = 0 and a fixed prime number p, the polynomial f (X) + cp k is irreducible over Q for all but finitely many positive integers k. If f (X) ∈ Z[X] has a rational root and c = 0 is a fixed integer, then f (X) + cp is irreducible over Q for all but finitely many prime numbers p. Inspired by some results of Fried [6] and Langmann [7] in connection with Hilbert's irreducibility theorem, Cavachi [2] studied the irreducibility of polynomials of the form f (X)+pg(X) with p prime and f , g relatively prime, and proved that for any relatively prime f, g ∈ Q[X] with deg f < deg g, the polynomial f (X) + pg(X) is irreducible over Q for all but finitely many prime numbers p. In [3] this result was strengthened, by providing an explicit bound α depending on f and g such that for all primes p > α the polynomial f (X) + pg(X) is irreducible over Q. Explicit upper bounds for the number of factors over Q of a linear combination n 1 f (X) + n 2 g(X), in particular irreducibility criteria covering also the case deg f = deg g, have been derived