Variations on a theorem of Capelli

Abstract

Elementary irreducibility criteria are established for [Formula: see text] where [Formula: see text] is irreducible over [Formula: see text] and [Formula: see text] is a prime. For instance, our main criterion implies that if [Formula: see text] is reducible over [Formula: see text], then [Formula: see text] divides [Formula: see text] modulo [Formula: see text]. Among several applications, it is shown that if [Formula: see text] has coefficients in [Formula: see text], then [Formula: see text] is irreducible over [Formula: see text] excluding a couple of obvious exceptions. As another application, it is proved that if [Formula: see text] and [Formula: see text] are distinct integers, then for [Formula: see text], the polynomial [Formula: see text] is irreducible over [Formula: see text] unless [Formula: see text] is odd and [Formula: see text]. Some emphasis is given to the non-cyclotomic monic polynomials [Formula: see text] with [Formula: see text]. In these cases, among other things, it is shown that if [Formula: see text], where [Formula: see text] denotes the height of [Formula: see text], then [Formula: see text] is irreducible over [Formula: see text]. Proofs of the irreducibility criteria rest upon a general result of Capelli concerning the factorization of [Formula: see text].