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

Masakazu Yamagishi

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