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The index of multiplicative groups of algebraic numbers. (English) Zbl 1141.11049

Sb. Math. 196, No. 9, 1307-1318 (2005); translation from Mat. Sb. 196, No. 9, 59-70 (2005).
Let \(K\) be a finite extension of \(\mathbb Q\) of degree \(D.\) Suppose that \(\kappa=1\) if \(K \subseteq\mathbb R\) and \(\kappa=2\) if \(K\) is complex. There are \(D\) embeddings of \(K\) into \(\mathbb C\), where \(\sigma_1\) is the identity, \(\sigma_2\) maps \(\alpha\) to its complex conjugate \(\overline{\alpha}\) if \(\alpha\) is complex, and the remaining embeddings are labelled in arbitrarily. For an algebraic number \(\alpha \in K^{*},\) set \(H^*(\alpha)=(2\kappa)^{-1}\sum_{\sigma>\kappa} | \log | \alpha| _{\sigma}| ,\) where \(\sigma\) is the index and \(| \alpha| _{\sigma}\) is the normalized valuation defined in a standard way so that \(| \alpha| _{\sigma}=| \sigma(\alpha)| \) for each \(1\leq \sigma \leq D\) and \(\sum_{\sigma=1}^{\infty}\log | \alpha| _{\sigma}=0\), where \(| \alpha| _{\sigma} \neq 1\) for only finitely many indices \(\sigma\). For a parameter \(E>0\), the author defines \(H_{E}^{*}(\alpha):=\max\{H^{*}(\alpha), E| \log \alpha| \}.\) Fix \(n\) nonzero numbers \(\alpha_1,\dots,\alpha_n\) of the field \(K\) whose logarithms are linearly independent over \(\mathbb Q\). Consider the numbers \(\beta \in K\) expressible in the form \(\alpha_1^{m_1} \dots \alpha_n^{m_n}\) with rational numbers \(m_1,\dots,m_n\). The set \(M\) of all such vectors \((m_1,\dots,m_n)\) is a lattice, \(\mathbb Z^n\) is its sublattice of a finite index. Set \(N=[M : \mathbb Z^n]\). In his main theorem, the author proves an estimate for \(N\) from above in terms of \(H^{*}(\alpha_1),\dots,H^{*}(\alpha_n),D\) and some other parameters which must be chosen to satisfy some inequalities. In the proof, he uses geometry of numbers. In the final section, the author explains how one can choose those additional parameters in order to satisfy his inequalities.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11H06 Lattices and convex bodies (number-theoretic aspects)
11J86 Linear forms in logarithms; Baker’s method
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