  In this paper, we study the critical orbit of a post-critically finite
polynomial of the form $f_{c,d}(x) = x^d+c \in \mathbb{C}[x]$. We discover that
in many cases the orbit elements satisfy some strong arithmetic properties. It
is well known that the $c$ values for which $f_{c,d}$ has tail size $m\geq 1$
and period $n$ are the roots of a polynomial $G_d(m,n) \in \mathbb{Z}[x]$, and
the irreducibility or not of $G_d(m,n)$ has been a great mystery. As a
consequence of our work, for any prime $d$, we establish the irreducibility of
these $G_d(m,n)$ polynomials for infinitely many pairs $(m,n)$. These appear to
be the first known such infinite families of $(m,n)$. We also prove that all
the iterates of $f_{c,d}$ are irreducible over $\mathbb{Q}(c)$ if $d$ is a
prime and $f_{c,d}$ has a fixed point in its post-critical orbit.
