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?