Function: nfresolvent
Section: number_fields
C-Name: nfresolvent
Prototype: GD0,L,
Help: nfresolvent(pol,{flag=0}): In the case where the Galois closure of the
 number field defined by pol is S3, Dl, A4, S4, F5, A5, M21, or M42, give the
 corresponding resolvent field. Otherwise, give a "canonical" subfield,
 or if flag >= 2 all "canonical" subfields. If flag is odd, give also the
 "conductor" f, whose definition is specific to each group.
Doc: Let \kbd{pol} be an irreducible integral polynomial defining a number
 field $K$ with Galois closure $\tilde{K}$. This function is limited to the
 Galois groups supported by \kbd{nflist}; in the following $\ell$ denotes an
 odd prime. If $\text{Gal}(\tilde{K}/\Q)$ is $D_\ell$, $A_4$, $S_4$, $F_5$
 ($M_{20}$), $A_5$, $M_{21}$ or $M_{42}$,
 return a polynomial $R$ defining the corresponding resolvent field (quadratic
 for $D_\ell$, cyclic cubic for $A_4$ and $M_{21}$, noncyclic cubic for $S_4$,
 cyclic quartic for $F_5$, $A_5(6)$ sextic for $A_5$, and cyclic sextic for
 $M_{42}$). In the $A_5(6)$ case, return the $A_5$ field of which it is the
 resolvent. Otherwise, give a ``canonical'' subfield, or $0$ if the Galois
 group is not supported.

 The binary digits of \fl\ correspond to 0: return a pair $[R,f]$ where $f$
 is a ``conductor'' whose definition is specific to each group and given
 below; 1: return all ``canonical'' subfields.

 Let $D$ be the discriminant of the resolvent field \kbd{nfdisc}$(R)$:

 \item In cases $C_\ell$, $D_\ell$, $A_4$, or $S_4$, $\text{disc}(K)
 =(Df^2)^m$ with $m=(\ell-1)/2$ in the first two cases, and $1$ in the last
 two.

 \item In cases where $K$ is abelian over the resolvent subfield, the conductor
 of the relative extension.

 \item In case $F_5$, $\text{disc}(K)=Df^4$ if $f>0$ or $5^2Df^4$ if $f<0$.

 \item In cases $M_{21}$ or $M_{42}$, $\text{disc}(K)=D^mf^6$ if $f>0$ or
 $7^3D^mf^6$ if $f<0$, where $m=2$ for $M_{21}$ and $m=1$ for $M_{42}$.

 \item In cases $A_5$ and $A_5(6)$, $\fl$ is currently ignored.
