Abstract: For a field $F$ containing a primitive $p$-root of $1$, let $G$ be the Galois group of the maximal $p$-extension $F(p)$ of $F$. The group $G$ might become very fat -- i.e., the size of its open subgroups might increase arbitrarily. This does not happen (namely, the size of its open subgroups is always the same) precisely when $F$ needs to eat only the roots of $p$-power index of its elements to reach $F(p)$. In this case we may compute explicitly the structure of $G$.