Heights of algebraic numbers modulo multiplicative group actions (Q1024405)

Let \(K\) be a number field, and let \(G\) be a subgroup of the multiplicative group of \(K\). Silverman defined the \(G\)-height of an algebraic number \(\theta\) by the formula \(\mathcal{H}(\theta,G):=\inf H(g^{1/n} \theta)\), where \(H\) is the usual absolute height and the infimum is taken over every \(g \in G\) and over every positive integer \(n\). Let \(K^1\) be the kernel of the norm map from \(K^*\) to \({\mathbb Q}^*\), and let \(E_K\) be the group of units of \(K\). Then \(\{1\} \subset E_K \subset K^1 \subset K^*\), and so \[ H(\theta) \geq \mathcal{H}(\theta,E_K) \geq \mathcal{H}(\theta, K^1) \geq \mathcal{H}(\theta,K^*). \] The formula involving a product over the archimedean places of \(K(\theta)\) for \(\mathcal{H}(\theta,E_K)\) was established by \textit{A.-M. Bergé} and \textit{J. Martinet} [Sémin. Théor. Nombres, Univ. Bordeaux I 1987/1988, Exp. No. 11, 28 p. (1988; Zbl 0699.12013)]. The authors find similar formulas for \(\mathcal{H}(\theta, K^1)\) and \(\mathcal{H}(\theta,K^*)\) and show that \(K^1\)-height (resp. \(K^*\)-height) is equal to \(1\) if and only if \(\theta^n \in K^1\) (resp. \(\theta^n \in K^*\)) for some positive integer \(n\).