LEHMER'S PROBLEM AND SPLITTING OF RATIONAL PRIMES IN NUMBER FIELDS

Let alpha be a non-zero algebraic integer of degree d which is not a root of unity. We prove that, if there exists an odd prime p with either (1) p <= d+ 1 and pO(Q(alpha)) = P1 P2 ... P-d, where P-1,..., P-d are distinct prime ideals of O-Q(alpha), or (2) p <= root d and pO(Q(alpha)) = P(1)(e1)P2(e2)... P-g(eg), where max(1 <= i <= g){ei} <= p and Sigma(g)(i=1) ei = d, then M(alpha) >= p/2. We also prove that if the residual degrees of primes in O-Q(alpha) which are lying above 2 are 1, then M(alpha) >= 2(1/4). This generalizes a result of Garza.