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

Yamagishi, Masakazu

Geodesic

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A note on free pro-

p

-extensions of algebraic number fields

Yamagishi, Masakazu

Journal de théorie des nombres de Bordeaux, Tome 5 (1993) no. 1, pp. 165-178

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Résumé

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 $1 \leq ρ \leq r_{2} + 1$ , ( $r_{2}$ 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.

MR Zbl

Keywords: algebraic number field, $\mathbb {Z}_p$-extension, free pro-$p$-group

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     title = {A note on free pro-$p$-extensions of algebraic number fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {165--178},
     publisher = {Universit\'e Bordeaux I},
     volume = {5},
     number = {1},
     year = {1993},
     mrnumber = {1251235},
     zbl = {0784.11052},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JTNB_1993__5_1_165_0/}
}

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Yamagishi, Masakazu. A note on free pro-$p$-extensions of algebraic number fields. Journal de théorie des nombres de Bordeaux, Tome 5 (1993) no. 1, pp. 165-178. http://geodesic.mathdoc.fr/item/JTNB_1993__5_1_165_0/