On metric heights
The naive height \(H(\alpha)\) of an algebraic number \(\alpha\) is defined to be the maximum of the absolute values of the coefficients of its minimal polynomial. Define the metric height of \(\alpha\) by \(\widehat{H}(\alpha) = \inf \{H(\alpha_1) \cdots H(\alpha_s)\}\), where the infimum is taken over all \(s \geq 1\) and all algebraic numbers \(\alpha_j\) such that \(\alpha_1 \cdots \alpha_s = \alpha\). The authors study the basic properties of the metric height and then show that \(\widehat{H}(m \sqrt{n}\,) = mn\) for natural numbers \(m, n\) with \(n > 1\) squarefree; they also manage to compute the metric height of other \(d\)-th roots of integers with few prime factors and observe that general formulas even for cubic surds seem to be difficult to obtain.
scientific article; zbMATH DE number 2099013