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Let $n$ be an odd prime and $l$ also a prime s.t. $l\equiv1 \bmod n$. I want a fast way to compute the $n^{th}$ degree subextension of the $l^{th}$ cyclotomic field. I need to compute lots of these in magma so the faster the better.

Ennola-Turunen's paper "On Cyclic Cubic Fields" enables me to do this very quickly for the $n=3$ case, but for the general case I am simply doing the following

  F:=CyclotomicField(l);
  G:=GaloisGroup(F);
  U:=sub<G|G.1^n>;
  f:=GaloisSubgroup(F,U);
  K:=NumberField(f);

which is slower than I'd like. It would be lovely if I could find something similar to help me compute these more quickly in the general case. Does anyone have any suggestions or references?

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