On a height related to the \(abc\) conjecture. (Q1432762)

Let \(a\) be an algebraic number, \(K\) an algebraic number field of degree \(d\), containing \(a\), let \(\Omega\) be the set of all normalized valuations of \(K\), and put \[ h_1(a)={1\over d}\sum_{v\in\Omega}\log(\max\{1,v(a),v(1-a)\}). \] The author shows, answering a question of Browkin, that for every \(X\) the set of all algebraic numbers \(a\) satisfying \([Q(a):Q]h_0(a)\leq X\) is finite.