Function: rnfkummer
Section: number_fields
C-Name: rnfkummer
Prototype: GDGD0,L,p
Help: rnfkummer(bnr,{subgp},{d=0}): bnr being as output by bnrinit,
finds a relative equation for the class field corresponding to the module in
bnr and the given congruence subgroup (the ray class field if subgp is
omitted). d can be zero (default), or positive, and in this case the
output is the list of all relative equations of degree d for the given bnr,
with the same conductor as (bnr, subgp).
Doc: \var{bnr}
being as output by \kbd{bnrinit}, finds a relative equation for the
class field corresponding to the module in \var{bnr} and the given
congruence subgroup (the full ray class field if \var{subgp} is omitted).
If $d$ is positive, outputs the list of all relative equations of
degree $d$ contained in the ray class field defined by \var{bnr}, with
the \emph{same} conductor as $(\var{bnr}, \var{subgp})$.
\misctitle{Warning} This routine only works for subgroups of prime index. It
uses Kummer theory, adjoining necessary roots of unity (it needs to compute a
tough \kbd{bnfinit} here), and finds a generator via Hecke's characterization
of ramification in Kummer extensions of prime degree. If your extension does
not have prime degree, for the time being, you have to split it by hand as a
tower / compositum of such extensions.