On a special case of Hilbert's irreducibility theorem (Q1587661)

Let \(K\) be a number field normal over \(\mathbb Q\), and let \(P\) be the set of rational primes that remain prime in \(K\). The author proves the following statement: Let \(f,g\in K[X]\) be two polynomials with no common root. If \(\text{deg }g>\text{deg }f\), then the polynomial \(f+pg\) is reducible for at most finitely many primes \(p\in P\). A similar statement is proved also for polynomials \(f,g\in K[X_1,\dots,X_n]\), under the condition \(\text{deg}_{X_1}g>\text{deg}_{X_1}f\).