Linearly independant Kummer extensions

[grenie@matapp.unimib.it (Loic Grenie)]

    Hi,

    I need to find the compositum of all Kummer extensions (of prime degree
  ell) of a base field F that do not lead to residual extensions (more
  precisely the residual extensions have predefined maximal size and F has
  already these residual dimensions). The condition is obviously linear in the
  ray class group, so that it would be enough to have a basis of linearly
  independant Kummer extensions of F.

    I theoretically know how to do it:

  I already have a modified version of rnfkummer that gives me a list of
  linearly independant extensions (I can submit it if you like, but it's lame);
  I can iterate over the subgroups matdiagonal(vector(#bnr.cyc,j,1+(j==i)))
  (1<=i<=#bnr.cyc), eliminating the subgroups with same conductor.

    Does anybody know a better method ? Should I start with certain
  conductors ? Should I randomly pick the extensions (including randomly
  picking the conductor) ?

    Thanks,

       Loïc