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 μ ⊂ k), we study the maximal rank ρ of a free pro-p- extension of k. We give various interpretations of 1 + r(k) - ρ. 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 - ρ is related to the torsion of a certain Iwasawa module, hence to the dualizing module of a certain Galois group (under Leopoldt's conjecture).