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On the \(v\)-adic independence of algebraic numbers. (English) Zbl 0788.11051

Gouvêa, Fernando Q. (ed.) et al., Advances in number theory. The proceedings of the third conference of the Canadian Number Theory Association, held at Queen’s University, Kingston, Canada, August 18-24, 1991. Oxford: Clarendon Press. 441-451 (1993).
Let \(v\) be a place of \(\mathbb{Q}\), \(U_ v\) the subgroup of \(\overline {\mathbb{Q}}_ v^*\) consisting of elements which are prime to \(v\) if \(v\) is finite, \(U_ v=\overline {\mathbb{Q}}_ v^*\) if \(v\) is infinite, \(M\) a finitely generated subgroup of \(U_ v\). The author defines the \(v\)-adic rank of \(M\) and states the following conjecture: if \(N\) denotes the subgroup of \(U_ v\) generated by the elements of \(M\) and their conjugates, if \(K\) is the extension of \(\mathbb{Q}\) generated by \(N\), and \(G\) the Galois group of \(K\) over \(\mathbb{Q}\), then the \(v\)-adic rank of \(M\) is the largest integer \(r\) for which there exists a \(\mathbb{Q}[G]\)-homomorphism \(\varphi\) from \(\mathbb{Q}\otimes_ \mathbb{Z} N\) to \(\mathbb{Q}[G]\) such that \(\dim_ \mathbb{Q} \varphi(\mathbb{Q}\otimes_ \mathbb{Z} M)=r\). This conjecture extends a conjecture of J.-F. Jaulent [J. Number Theory 20, 149-158 (1985; Zbl 0571.12007)], which deals only with the special case where \(M\) is stable under the action of the Galois group of \(\overline{\mathbb{Q}}\) over \(\mathbb{Q}\); hence the author’s conjecture contains Leopoldt’s conjecture on the \(p\)-adic rank of the units of an algebraic number field (nonvanishing of the \(p\)-adic regulator).
The author shows that his conjecture is a consequence of the algebraic independence of linearly independent (\(v\)-adic) logarithms of algebraic numbers. Finally, he provides partial solutions, and in particular he proves his conjecture in certain cases.
For the entire collection see [Zbl 0773.00021].

MSC:

11R27 Units and factorization
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11J81 Transcendence (general theory)

Citations:

Zbl 0571.12007
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