  Let $t$ and $x$ be indeterminates, let $\phi(x)=x^2+t\in\mathbb Q(t)[x]$, and
for every positive integer $n$ let $\Phi_n(t,x)$ denote the $n^{\text{th}}$
dynatomic polynomial of $\phi$. Let $G_n$ be the Galois group of $\Phi_n$ over
the function field $\mathbb Q(t)$, and for $c\in\mathbb Q$ let $G_{n,c}$ be the
Galois group of the specialized polynomial $\Phi_n(c,x)$. It follows from
Hilbert's irreducibility theorem that for fixed $n$ we have $G_n\cong G_{n,c}$
for every $c$ outside a thin set $E_n\subset\mathbb Q$. By earlier work of
Morton (for $n=3$) and the present author (for $n=4$), it is known that $E_n$
is infinite if $n\le 4$. In contrast, we show here that $E_n$ is finite if
$n\in\{5,6,7,9\}$. As an application of this result we show that, for these
values of $n$, the following holds with at most finitely many exceptions: for
every $c\in\mathbb Q$, more than $81\%$ of prime numbers $p$ have the property
that the polynomial $x^2+c$ does not have a point of period $n$ in the $p$-adic
field $\mathbb Q_p$.
