Function: idealfrobenius
Section: number_fields
C-Name: idealfrobenius
Prototype: GGG
Help: idealfrobenius(nf,gal,pr): returns the Frobenius element (pr|nf/Q)
attached to the unramified prime ideal pr in prid format, in the Galois
group gal of the number field nf.
Doc: Let $K$ be the number field defined by $nf$ and assume $K/\Q$ be a
Galois extension with Galois group given \kbd{gal=galoisinit(nf)},
and that \var{pr} is an unramified prime ideal $\goth{p}$ in \kbd{prid}
format.
This function returns a permutation of \kbd{gal.group} which defines
the Frobenius element $\Frob_{\goth{p}}$ attached to $\goth{p}$.
If $p$ is the unique prime number in $\goth{p}$, then
$\Frob(x)\equiv x^p\mod\goth{p}$ for all $x\in\Z_K$.
\bprog
? nf = nfinit(polcyclo(31));
? gal = galoisinit(nf);
? pr = idealprimedec(nf,101)[1];
? g = idealfrobenius(nf,gal,pr);
? galoispermtopol(gal,g)
%5 = x^8
@eprog\noindent This is correct since $101\equiv 8\mod{31}$.