We describe several algorithms for computing e-th roots of elements in a number field K, where e is an odd prime integer. In particular we generalize Couveignes’ and Thomé’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.