The goal of this paper is to obtain lower bounds on the height of an
algebraic number in a relative setting, extending previous work of Amoroso and
Masser. Specifically, in our first theorem we obtain an effective bound for the
height of an algebraic number α when the base field K is a
number field and K(α)/K is Galois. Our second result
establishes an explicit height bound for any non-zero element α which is
not a root of unity in a Galois extension F/K, depending on
the degree of K/Q and the number of conjugates of α
which are multiplicatively independent over K. As a consequence, we
obtain a height bound for such α that is independent of the
multiplicative independence condition