11.85 (1990)

Let $F$ be a free pro-$p$-group with a basis $X$ and let $R = r^F$ be the closed normal subgroup generated by an element $r$. We say that an element $s$ of $F$ is associated to $r$ if $s^F = R$.
$\qquad$ a) Suppose that none of the elements associated to $r$ is a $p$-th power of an element in $F$. Is it true that $F/R$ is torsion-free?
$\qquad$ b) Among the elements associated to $r$ there is one that depends on the minimal subset $X'$ of the basis $X$. Let $x \in X'$. Is it true that the images of the elements of $X \setminus \{x\}$ in $F/R$ freely generate a free pro-$p$-group?