Let α1, …, αn an be non-zero algebraic numbers with degrees at most d and heights respectively Al, …, An (all Aj ≥ 4) and let b1, …, bn be rational integers with absolute values at most B (≥ 4). Denote by p a prime ideal of the field and suppose that p divides the rational prime p. Write
Then it is shown that
for some effectively computable constant C > 0 depending only on n, d and p. The argument suffices to prove similarly that in the complex case, if
for any fixed determination of the logarithms, then
for some effectively computable constant C′ > 0 depending only on n and d (and he determination of the logarithms).