Function: nfhilbert
Section: number_fields
C-Name: nfhilbert0
Prototype: lGGGDG
Help: nfhilbert(nf,a,b,{pr}): if pr is omitted, global Hilbert symbol (a,b) in
 nf, that is 1 if X^2-aY^2-bZ^2 has a nontrivial solution (X,Y,Z) in nf, -1
 otherwise. Otherwise compute the local symbol modulo the prime ideal pr.
Doc: if \var{pr} is omitted,
 compute the global quadratic \idx{Hilbert symbol} $(a,b)$ in $\var{nf}$, that
 is $1$ if $x^{2} - a y^{2} - b z^{2}$ has a non trivial solution $(x,y,z)$ in
 $\var{nf}$, and $-1$ otherwise. Otherwise compute the local symbol modulo
 the prime ideal \var{pr}, as output by \kbd{idealprimedec}.

 The quaternion algebra~$(a,b)$ over \var{nf} can be created with
 \kbd{alginit(nf,[a,b])} (see \tet{alginit}).
Variant:
 Also available is \fun{long}{nfhilbert}{GEN nf,GEN a,GEN b} (global
 quadratic Hilbert symbol), where \kbd{nf} is a true \var{nf} structure.


