bnflogdegree

Function: bnflogdegree
Section: number_fields
C-Name: bnflogdegree
Prototype: GGG
Help: bnflogdegree(nf, A, l): let A be an ideal, return exp(deg_F A)
 the exponential of the l-adic logarithmic degree.
Doc: Let \var{nf} be a \var{nf} structure attached to a number field $F$,
 and let $l$ be a prime number (hereafter
 denoted $\ell$). The
 $\ell$-adified group of id\`{e}les of $F$ quotiented by
 the group of logarithmic units is identified to the $\ell$-group
 of logarithmic divisors $\oplus \Z_\ell [\goth{p}]$, generated by the
 maximal ideals of $F$.

 The \emph{degree} map $\deg_F$ is additive with values in $\Z_\ell$,
 defined by $\deg_F \goth{p} = \tilde{f}_{\goth{p}} \deg_\ell p$,
 where the integer $\tilde{f}_{\goth{p}}$ is as in \tet{bnflogef} and $\deg_\ell p$
 is $\log_\ell p$ for $p\neq \ell$, $\log_\ell (1 + \ell)$ for
 $p = \ell\neq 2$ and $\log_\ell (1 + 2^2)$ for $p = \ell = 2$.

 Let $A = \prod \goth{p}^{n_{\goth{p}}}$ be an ideal and let $\tilde{A} =
 \sum n_\goth{p} [\goth{p}]$ be the attached logarithmic divisor. Return the
 exponential of the $\ell$-adic logarithmic degree $\deg_F A$, which is a
 natural number.