Function: idealfrobenius
Section: number_fields
C-Name: idealfrobenius
Prototype: GGG
Help: idealfrobenius(nf,gal,pr): Returns the Frobenius element (pr|nf/Q)
 associated with 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 $pr$ is the prime ideal $\goth{P}$ in prid format, and that
 $\goth{P}$ is unramified.
 This function returns a permutation of \kbd{gal.group} which defines the
 automorphism $\sigma=\left(\goth{P}\over K/\Q \right)$, i.e the Frobenius
 element associated to $\goth{P}$. If $p$ is the unique prime number
 in $\goth{P}$, then $\sigma(x)\equiv x^p\mod\P$ 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 $101\equiv 8\mod{31}$.