  We prove the irreducibility of integer polynomials $f(X)$ whose roots lie
inside an Apollonius circle associated to two points on the real axis with
integer abscisae $a$ and $b$, with ratio of the distances to these points
depending on the canonical decomposition of $f(a)$ and $f(b)$. In particular,
we obtain irreducibility criteria for the case where $f(a)$ and $f(b)$ have few
prime factors, and $f$ is either an Enestr\"om-Kakeya polynomial, or has a
large leading coefficient. Analogous results are also provided for multivariate
polynomials over arbitrary fields, in a non-Archimedean setting.
