Testing equalities of multiplicative representations in polynomial time

For multiplicative representations Π_i=1^kα_iⁿ⁽ⁱ⁾ and Π_j=1^lβ_j^m(j) where α_i, β_j are non-zero elements of some algebraic number field K and n_i, m_j are rational integers, we present a deterministic polynomial time algorithm that decides whether Π_i=1^kα_iⁿ⁽ⁱ⁾ equals Π_j=1^lβ_j^m(j). The running time of the algorithm is polynomial in the number of bits required to represent the number field K, the elements α_i , β_j and the integers n_i, m_j