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The answer is yes. , assuming that the two-variable polynomial $f_n(x)T^n + \dots + f_1(x)T + f_0(x)$ is irreducible over $K$.

This follows from the version of Hilbert's irreducibility theorem for number fields proved as Theorem 46 of p.298 of Schinzel's book Polynomials with special regard to reducibility: the relevant passage can be viewed on Google Books

In fact, if I'm reading it correctly, it looks like one has irreducibility for all rational integers $\alpha$ belonging to an appropriate residue class.
In fact, if I'm reading it correctly, it looks like one has irreducibility for all rational integers $\alpha$ belonging to an appropriate residue class.