Finite-quotient criterion for roots in a profinite group
= Finite-quotient criterion for roots in a profinite group
For $G=\varprojlim_jG_j$, the equation $x^m=g$ has a solution in $G$ exactly when it has a solution after projection to every $G_j$. The finite nonempty sets of roots form an inverse system, and a compatible family is a root in $G$.