  Given a separable nonconstant polynomial $f(x)$ with integer coefficients, we
consider the set $S$ consisting of the squarefree parts of all the rational
values of $f(x)$, and study its behavior modulo primes. Fixing a prime $p$, we
determine necessary and sufficient conditions for $S$ to contain an element
divisible by $p$. Furthermore, we conjecture that if $p$ is large enough, then
$S$ contains infinitely many representatives from every nonzero residue class
modulo $p$. The conjecture is proved by elementary means assuming $f(x)$ has
degree 1 or 2. If $f(x)$ has degree 3, or if it has degree 4 and has a rational
root, the conjecture is shown to follow from the Parity Conjecture for elliptic
curves. For polynomials of arbitrary degree, a local analogue of the conjecture
is proved using standard results from class field theory, and empirical
evidence is given to support the global version of the conjecture.
