  Fix an odd prime $p$, and let $F$ be a field containing a primitive $p$th
root of unity. It is known that a $p$-rigid field $F$ is characterized by the
property that the Galois group $G_F(p)$ of the maximal $p$-extension $F(p)/F$
is a solvable group. We give a new characterization of $p$-rigidity which says
that a field $F$ is $p$-rigid precisely when two fundamental canonical
quotients of the absolute Galois groups coincide. This condition is further
related to analytic $p$-adic groups and to some Galois modules. When $F$ is
$p$-rigid, we also show that it is possible to solve for the roots of any
irreducible polynomials in $F[X]$ whose splitting field over $F$ has a
$p$-power degree via non-nested radicals. We provide new direct proofs for
hereditary $p$-rigidity, together with some characterizations for $G_F(p)$ --
including a complete description for such a group and for the action of it on
$F(p)$ -- in the case $F$ is $p$-rigid.
