Variations on a theorem of Capelli

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.