Function: issquare
Section: number_theoretical
C-Name: issquareall
Prototype: lGD&
Help: issquare(x,{&n}): true(1) if x is a square, false(0) if not. If n is
 given puts the exact square root there if it was computed.
Description:
 (int):bool        Z_issquare($1)
 (gen):bool        issquare($1)
 (int, &int):bool  Z_issquareall($1, &$2)
 (gen, &gen):bool  issquareall($1, &$2)
Doc: true (1) if $x$ is a square, false (0)
 if not. What ``being a square'' means depends on the type of $x$: all
 \typ{COMPLEX} are squares, as well as all nonnegative \typ{REAL}; for
 exact types such as \typ{INT}, \typ{FRAC} and \typ{INTMOD}, squares are
 numbers of the form $s^{2}$ with $s$ in $\Z$, $\Q$ and $\Z/N\Z$ respectively.
 \bprog
 ? issquare(3)          \\ as an integer
 %1 = 0
 ? issquare(3.)         \\ as a real number
 %2 = 1
 ? issquare(Mod(7, 8))  \\ in Z/8Z
 %3 = 0
 ? issquare( 5 + O(13^4) )  \\ in Q_13
 %4 = 0
 @eprog
 If $n$ is given, a square root of $x$ is put into $n$.
 \bprog
 ? issquare(4, &n)
 %1 = 1
 ? n
 %2 = 2
 @eprog
 For polynomials, either we detect that the characteristic is 2 (and check
 directly odd and even-power monomials) or we assume that $2$ is invertible
 and check whether squaring the truncated power series for the square root
 yields the original input.

 For \typ{POLMOD} $x$, we only support \typ{POLMOD}s of \typ{INTMOD}s
 encoding finite fields, assuming without checking that the intmod modulus
 $p$ is prime and that the polmod modulus is irreducible modulo $p$.
 \bprog
 ? issquare(Mod(Mod(2,3), x^2+1), &n)
 %1 = 1
 ? n
 %2 = Mod(Mod(2, 3)*x, Mod(1, 3)*x^2 + Mod(1, 3))
 @eprog

Variant: Also available is \fun{long}{issquare}{GEN x}. Deprecated
 GP-specific functions \fun{GEN}{gissquare}{GEN x} and
 \fun{GEN}{gissquareall}{GEN x, GEN *pt} return \kbd{gen\_0} and \kbd{gen\_1}
 instead of a boolean value.
