  For a real polynomial $f$ we present explicit zero-free angular sectors in
the complex plane, symmetric with respect to the real axis, with angles
depending only on the degree of $f$, and vertices expressed in terms of the
coefficients of $f$. We also describe zero-free lens-shaped regions for $f$
that are associated to the zero-free sectors of the reciprocal of $f$. As an
application, we use these zero-free regions to obtain irreducibility criteria
for polynomials with integer coefficients that take a prime or a prime power
value.
