Function: nfeltsign
Section: number_fields
C-Name: nfeltsign
Prototype: GGDG
Help: nfeltsign(nf,x,{pl}): signs of real embeddings of x at places given
 by vector pl.
Doc: given an element $x$ in the number field \var{nf}, returns the signs of
 the real embeddings of $x$ specified by optional argument \var{pl}:

 \item \var{pl} omitted: return the vector of signs at all $r_{1}$ real places;

 \item \var{pl} an integer between $1$ and $r_{1}$: return the sign of the
 $i$-th embedding of $x$, attached to the $i$-th real root of \kbd{nf.pol},
 i.e. \kbd{nf.roots$[i]$};

 \item \var{pl} a vector or \typ{VECSMALL}: return the vector of signs; the $i$-th
 entry gives the sign at the real place attached to the $\var{pl}[i]$-th real
 root of \kbd{nf.pol}.

 \bprog
 ? nf = nfinit(polsubcyclo(11,5,'y)); \\ Q(cos(2 pi/11))
 ? nf.sign
 %2 = [5, 0]
 ? x = Mod('y, nf.pol);
 ? nfeltsign(nf, x)
 %4 = [-1, -1, -1, 1, 1]
 ? nfeltsign(nf, x, 1)
 %5 = -1
 ? nfeltsign(nf, x, [1..4])
 %6 = [-1, -1, -1, 1]
 ? nfeltsign(nf, x, 6) \\ there are only 5 real embeddings
  ***   at top-level: nfeltsign(nf,x,6)
  ***                 ^-----------------
  *** nfeltsign: domain error in nfeltsign: index > 5
 @eprog
