padic_sqrt

A nonzero $x = p^v u$ is a square if and only if $v$ is even and the unit $u$ is a square in $\mathbf{Z}_p$; for odd $p$ this holds if and only if $u$ is a square modulo $p$, and for $p = 2$ if and only if $u \equiv 1 \pmod 8$. The root is $p^{v/2}$ times a square root of $u$ modulo $p^{N - v/2}$, obtained from a root modulo $p$ (or the congruence modulo 8) by Hensel lifting of the inverse square root (Hensel lifting of inverses, square roots and Teichmüller lifts). The decision is made from the exact value of op even when the precision of rop is too small to hold a nonzero root.