  For a degree $n$ polynomial $f$ over the rationals, the elements in the fiber
$f^{-1}(a)$ are of degree $n$ over $\mathbb Q$ for most rational values $a$ by
Hilbert's irreducibility theorem. Determining the set of exceptional $a$'s
without this property is a long standing open problem that is closely related
to the Davenport--Lewis--Schinzel problem (1959) on reducibility of separated
polynomials. As opposed to previous work which mostly concerns indecomposable
$f$, we answer both problems for decomposable $f=f_1\circ\cdots\circ f_r$, as
long as the indecomposable factors $f_i\in\mathbb Q[x]$ are of degree at least
$5$ and are not $x^n$ or a Chebyshev polynomial composed with linear
polynomials.
