On a Special Case of Hilbert's Irreducibility Theorem

Abstract

We prove that if K is a finite extension of Q, P is the set of prime numbers in Z that remain prime in the ring R of integers of K, f, g∈K[X] with degg>degf and f, g are relatively prime, then f+pg is reducible in K[X] for at most a finite number of primes p∈P. We then extend this property to polynomials in more than one indeterminate. These results are related to Hilbert's irreducibility theorem.