Roots of unity as quotients of two conjugate algebraic numbers

Sa？etak Let α be an algebraic number of degree d ≥ 2 over Q. Suppose for some pairwise coprime positive integers n1,… ,nr we have deg(αnj) < d for j=1,…,r, where deg(αn)=d for each positive proper divisor n of nj. We prove that then φ(n1 … nr) ≤ d, where φ stands for the Euler totient function. In particular, if nj=pj, j=1,…,r, are any r distinct primes satisfying deg(αpj) < d, then the inequality (p1-1)… (pr-1) ≤ d holds, and therefore r ？ log d/log log d for d ≥ 3. This bound on r improves that of Dobrowolski r ≤ log d/log 2 proved in 1979 and is best possible