Function: sqrtn
Section: transcendental
C-Name: gsqrtn
Prototype: GGD&p
Help: sqrtn(x,n,{&z}): nth-root of x, n must be integer. If present, z is
 set to a suitable root of unity to recover all solutions. If it was not
 possible, z is set to zero.
Doc: principal branch of the $n$th root of $x$,
 i.e.~such that $\text{Arg}(\text{sqrt}(x))\in{} ]-\pi/n, \pi/n]$. Intmod
 a prime and $p$-adics are allowed as arguments.

 If $z$ is present, it is set to a suitable root of unity allowing to
 recover all the other roots. If it was not possible, z is
 set to zero. In the case this argument is present and no square root exist,
 $0$ is returned instead or raising an error.
 \bprog
 ? sqrtn(Mod(2,7), 2)
 %1 = Mod(4, 7)
 ? sqrtn(Mod(2,7), 2, &z); z
 %2 = Mod(6, 7)
 ? sqrtn(Mod(2,7), 3)
   ***   at top-level: sqrtn(Mod(2,7),3)
   ***                 ^-----------------
   *** sqrtn: nth-root does not exist in gsqrtn.
 ? sqrtn(Mod(2,7), 3,  &z)
 %2 = 0
 ? z
 %3 = 0
 @eprog

 The following script computes all roots in all possible cases:
 \bprog
 sqrtnall(x,n)=
 { my(V,r,z,r2);
   r = sqrtn(x,n, &z);
   if (!z, error("Impossible case in sqrtn"));
   if (type(x) == "t_INTMOD" || type(x)=="t_PADIC",
     r2 = r*z; n = 1;
     while (r2!=r, r2*=z;n++));
   V = vector(n); V[1] = r;
   for(i=2, n, V[i] = V[i-1]*z);
   V
 }
 addhelp(sqrtnall,"sqrtnall(x,n):compute the vector of nth-roots of x");
 @eprog\noindent
Variant: If $x$ is a \typ{PADIC}, the function
 \fun{GEN}{Qp_sqrt}{GEN x, GEN n, GEN *z} is also available.