  We describe several algorithms for computing $e$-th roots of elements in a
number field $K$, where $e$ is an odd prime-power integer. In particular we
generalize Couveignes' and Thom\'e's algorithms originally designed to compute
square-roots in the Number Field Sieve algorithm for integer factorization. Our
algorithms cover most cases of $e$ and $K$ and allow to obtain reasonable
timings even for large degree number fields and large exponents $e$. The
complexity of our algorithms is better than general root finding algorithms and
our implementation compared well in performance to these algorithms implemented
in well-known computer algebra softwares. One important application of our
algorithms is to compute the saturation phase in the Twisted-PHS algorithm for
computing the Ideal-SVP problem over cyclotomic fields in post-quantum
cryptography.
