Function: algsplittingdata
Section: algebras
C-Name: algsplittingdata
Prototype: mG
Help: algsplittingdata(al): data stored in the central simple algebra al to
compute a splitting of al over an extension.
Doc: Given a central simple algebra \var{al} output by \tet{alginit} defined
by a multiplication table over its center~$K$ (a number field), returns data
stored to compute a splitting of \var{al} over an extension. This data is a
\typ{VEC} \kbd{[t,Lbas,Lbasinv]} with $3$ components:
\item an element $t$ of \var{al} such that $L=K(t)$ is a maximal subfield
of \var{al};
\item a matrix \kbd{Lbas} expressing a $L$-basis of \var{al} (given an
$L$-vector space structure by multiplication on the right) on the integral
basis of \var{al};
\item a matrix \kbd{Lbasinv} expressing the integral basis of \var{al} on
the previous $L$-basis.
\bprog
? nf = nfinit(y^3-5); a = y; b = y^2;
? {m_i = [0,a,0,0;
1,0,0,0;
0,0,0,a;
0,0,1,0];}
? {m_j = [0, 0,b, 0;
0, 0,0,-b;
1, 0,0, 0;
0,-1,0, 0];}
? {m_k = [0, 0,0,-a*b;
0, 0,b, 0;
0,-a,0, 0;
1, 0,0, 0];}
? mt = [matid(4), m_i, m_j, m_k];
? A = alginit(nf,mt,'x);
? [t,Lb,Lbi] = algsplittingdata(A);
? t
%8 = [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]~;
? matsize(Lb)
%9 = [12, 2]
? matsize(Lbi)
%10 = [2, 12]
@eprog