### Some classes of irreducible polynomials

Anca Iuliana Bonciocat, Nicolae Ciprian Bonciocat
2006 Acta Arithmetica
1. Introduction. Lipka  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  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  : 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  and Langmann  in connection with Hilbert's irreducibility theorem, Cavachi  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  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