On the Irreducibility of Polynomials with Leading Coefficient Divisible by a Large Prime Power

Abstract

aoan t? 0, p a prime number, and k a positive integer. For instance, he proved that for fixed p, ao, ax , . . . , an with a§a'an ^ 0, / will be irreducible over Qfor all but finitely many positive integers k. Similar results for the case a' = 0 and aoa2an ^ 0, and for polynomials in several variables over a given field, have been obtained in [1]. For several elegant connections between prime numbers and irreducible polynomials, the reader is referred to [4] and [2]. In this note we give an effective proof of Lipka's result, formulated for Xdeg/ f(l/X) (the reciprocal of /), thus providing explicit conditions on the leading coefficient which will ensure the irreducibility of polynomials of this type. More precisely, we prove the following: