Power detection over number fields

Abstract

We describe an algorithm that, given an integral ideal A \mathfrak {A} in a number field  K K , finds the largest integer k ⩾ 1 k\geqslant 1 such that A \mathfrak {A} is a k k -th power, and at the same time computes the ideal B \mathfrak {B} such that A = B k \mathfrak {A} = \mathfrak {B}^k . This algorithm does not require the complete factorization of A \mathfrak {A} into a product of prime ideals; given the maximal order Z K \mathbb {Z}_K , we prove that it has polynomial time complexity. We apply this algorithm to the reduction of elements of K ∗ K^* modulo k k -th powers.