Sur le rang d'une extension pro-p-libre d'un corps de nombres

For an algebraic number field $k$ and a prime number $p$ (if $p=2$, we assume that $\mu_4\subset k$), we study the maximal rank $\rho_k$ of a free pro-$p$-extension of $k$. We give various interpretations of $1+r_2(k)-\rho_k$. The first uses Iwasawa theory, the second uses the envelope of a module and the third is local-global. These expressions confirm that $1+r_2-\rho_k$ is related to the torsion of a certain Iwasawa module, hence to the dualizing module of a certain Galois group (under Leopoldt's conjecture).