On a special case of Hilbert's irreducibility theorem

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 is an element of K[X] with deg g > deg f and f, g are relatively prime, then f + pg is reducible in K[X] for at most a finite number of primes p is an element of P. We then extend this property to polynomials in more than one indeterminate. These results are related to Hilbert's irreducibility theorem. (C) 2000 Academic Press.