A note on free pro- p -extensions of algebraic number fields

Masakazu Yamagishi

Journal de théorie des nombres de Bordeaux (1993)

  • Volume: 5, Issue: 1, page 165-178
  • ISSN: 1246-7405

Abstract

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For an algebraic number field k and a prime p , define the number ρ 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 ≤ ρ ≤ r 2 + 1 , ( r 2 denotes the number of complex places of k ). Some examples of k and p with ρ = r 2 + 1 have been known so far. In this note, the invariant ρ is studied, and among other things some examples with ρ < r 2 + 1 are given.

How to cite

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Yamagishi, Masakazu. "A note on free pro-$p$-extensions of algebraic number fields." Journal de théorie des nombres de Bordeaux 5.1 (1993): 165-178. <http://eudml.org/doc/93570>.

@article{Yamagishi1993,
abstract = {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 $\rho &lt; r_2 + 1$ are given.},
author = {Yamagishi, Masakazu},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {algebraic number field; $\mathbb \{Z\}_p$-extension; free pro-$p$-group; weak Leopoldt conjecture; maximal rank; free pro- Galois groups},
language = {eng},
number = {1},
pages = {165-178},
publisher = {Université Bordeaux I},
title = {A note on free pro-$p$-extensions of algebraic number fields},
url = {http://eudml.org/doc/93570},
volume = {5},
year = {1993},
}

TY - JOUR
AU - Yamagishi, Masakazu
TI - A note on free pro-$p$-extensions of algebraic number fields
JO - Journal de théorie des nombres de Bordeaux
PY - 1993
PB - Université Bordeaux I
VL - 5
IS - 1
SP - 165
EP - 178
AB - 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 $\rho &lt; r_2 + 1$ are given.
LA - eng
KW - algebraic number field; $\mathbb {Z}_p$-extension; free pro-$p$-group; weak Leopoldt conjecture; maximal rank; free pro- Galois groups
UR - http://eudml.org/doc/93570
ER -

References

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