Function: bnfsignunit
Section: number_fields
C-Name: signunits
Prototype: G
Help: bnfsignunit(bnf): matrix of signs of the real embeddings of the system
 of fundamental units found by bnfinit.
Doc: $\var{bnf}$ being as output by
 \kbd{bnfinit}, this computes an $r_1\times(r_1+r_2-1)$ matrix having $\pm1$
 components, giving the signs of the real embeddings of the fundamental units.
 The following functions compute generators for the totally positive units:

 \bprog
 /* exponents of totally positive units generators on bnf.tufu */
 tpuexpo(bnf)=
 { my(S,d,K);

   S = bnfsignunit(bnf); d = matsize(S);
   S = matrix(d[1],d[2], i,j, if (S[i,j] < 0, 1,0));
   S = concat(vectorv(d[1],i,1), S);   \\ add sign(-1)
   K = lift(matker(S * Mod(1,2)));
   if (K, mathnfmodid(K, 2), 2*matid(d[1]))
 }

 /* totally positive units */
 tpu(bnf)=
 { my(vu = bnf.tufu, ex = tpuexpo(bnf));

   vector(#ex-1, i, factorback(vu, ex[,i+1]))  \\ ex[,1] is 1
 }
 @eprog