On a Special Case of Hilbertʹs Irreducibility Theorem Original Research Article

We prove that if K is a finite extension of image, P is the set of prime numbers in image that remain prime in the ring R of integers of K, f, gset membership, variantK[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 pset membership, variantP. We then extend this property to polynomials in more than one indeterminate. These results are related to Hilbertʹs irreducibility theorem.