  Let $p$ be an odd prime number and $F$ a field containing a primitive $p$th
root of unity. We prove a new restriction on the group-theoretic structure of
the absolute Galois group $G_F$ of $F$. Namely, the third subgroup $G_F^{(3)}$
in the descending $p$-central sequence of $G_F$ is the intersection of all open
normal subgroups $N$ such that $G_F/N$ is 1, $\mathbb{Z}/p^2$, or the modular
group $M_{p^3}$ of order $p^3$.
