bnflogef

Function: bnflogef
Section: number_fields
C-Name: bnflogef
Prototype: GG
Help: bnflogef(nf,pr): return [e~, f~] the logarithmic ramification and
 residue degrees for the maximal ideal pr.
Doc: let \var{nf} be a \var{nf} structure attached to a number field $F$
 and let \var{pr} be a \var{prid} structure attached to a
 maximal ideal $\goth{p} / p$. Return
 $[\tilde{e}(F_\goth{p} / \Q_p), \tilde{f}(F_\goth{p} / \Q_p)]$
 the logarithmic ramification and residue degrees. Let $\Q_p^c/\Q_p$ be the
 cyclotomic $\Z_p$-extension, then
 $\tilde{e} = [F_\goth{p} \colon F_\goth{p} \cap \Q_p^c]$ and
 $\tilde{f} = [F_\goth{p} \cap \Q_p^c \colon \Q_p]$. Note that
 $\tilde{e}\tilde{f} = e(\goth{p}/p) f(\goth{p}/p)$, where $e(\goth{p}/p)$ and $f(\goth{p}/p)$ denote the
 usual ramification and residue degrees.
 \bprog
 ? F = nfinit(y^6 - 3*y^5 + 5*y^3 - 3*y + 1);
 ? bnflogef(F, idealprimedec(F,2)[1])
 %2 = [6, 1]
 ? bnflogef(F, idealprimedec(F,5)[1])
 %3 = [1, 2]
 @eprog