Testing equalities of multiplicative representations in polynomial time

For multiplicative representations /spl Pi//sub i=1//sup k//spl alpha//sub i//sup n(i)/ and /spl Pi//sub j=1//sup l//spl beta//sub j//sup m(j)/ where /spl alpha//sub i/, /spl beta//sub j/ are non-zero elements of some algebraic number field K and n/sub i/, m/sub j/ are rational integers, we present a deterministic polynomial time algorithm that decides whether /spl Pi//sub i=1//sup k//spl alpha//sub i//sup n(i)/ equals /spl Pi//sub j=1//sup l//spl beta//sub j//sup m(j)/. The running time of the algorithm is polynomial in the number of bits required to represent the number field K, the elements /spl alpha//sub i/, /spl beta//sub j/ and the integers n/sub i/, m/sub j/.<>