My question is whether for every extension of number fields $L\subset K$, and for every $f_0(x),...,f_n(x)$ in $K[x]$, there is some $\alpha\in L$ such that $$f_n(\alpha)T^n+...+f_1(\alpha)T+f_0(\alpha)$$ is irreducible as a polynomial in $K[T]$.
If $L=K$ this is known from Hilbert's Irreducibility Theorem. I find it hard to believe that there is a counter-example to this, but on the other hand I can't seem to conjure up a proof.
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
http://books.google.ca/books?id=kca0JqBhnsIC&lpg=PA298&vq=hilbert's%20irreducibility%20theorem&pg=PA298#v=onepage&q&f=false
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.
