Height of algebraic units under splitting conditions.

Let α be a non-zero algebraic unit which is not a root of unity and K be a number field of degree d over Q . In this paper, we prove the following: Let P be a prime ideal of O K which lies above a rational odd prime p such that P O K (α) = P 1 e 1 P 2 e 2 ⋯ P g e g , where max 1 ≤ i ≤ g { e i } ≤ p and e 1 + ⋯ + e g = [ K (α) : K ]. Then h (α) ≥ c , where c > 0 is an effectively computable constant depending only on p and [ K : Q ] = d. This generalizes a result of Petsche. Also, we prove the following: Let P be a prime ideal of O K which lies above 2 such that P O K (α) = P 1 e 1 P 2 e 2 ⋯ P r e r , where e 1 + ⋯ + e r = [ K (α) : K ]. Then M (α) ≥ C (K) , where C (K) > 1 is a constant which depends only on [ K : Q ] = d. [ABSTRACT FROM AUTHOR]