### On free constructions.

Karl Strambach, Martin Funk (1991)

Manuscripta mathematica

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Karl Strambach, Martin Funk (1991)

Manuscripta mathematica

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F.-V. Kuhlmann, M. Pank, P. Roquette (1986)

Manuscripta mathematica

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Antonio José Engler (1995)

Manuscripta mathematica

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Arnaldo Garcia, Henning Sitchtenoth (1991)

Manuscripta mathematica

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Serban A. Basarab, F.-V. Kuhlmann (1992)

Manuscripta mathematica

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Serban A. Basarab, F.-V. Kuhlmann (1994)

Manuscripta mathematica

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Claus Scheiderer (1991)

Manuscripta mathematica

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Jochen Koenigsmann (1995)

Manuscripta mathematica

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Moshe Jarden (1988)

Manuscripta mathematica

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R. Gold, H. Kisilevsky (1988)

Manuscripta mathematica

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Antonio Leaci (1992)

Manuscripta mathematica

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Mamoru Asada (1985)

Mathematische Annalen

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Masakazu Yamagishi (1993)

Journal de théorie des nombres de Bordeaux

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For an algebraic number field $k$ and a prime $p$, define the number $\rho $ to be the maximal number $d$ such that there exists a Galois extension of $k$ whose Galois group is a free pro-$p$-group of rank $d$. The Leopoldt conjecture implies $1\le \rho \le {r}_{2}+1$, (${r}_{2}$ denotes the number of complex places of $k$). Some examples of $k$ and $p$ with $\rho ={r}_{2}+1$ have been known so far. In this note, the invariant $\rho $ is studied, and among other things some examples with $\rho \<{r}_{2}+1$ are given.