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 µ4 [contained in] k), we study the maximal rank [rho]k of a free pro-p-extension of k. We give various interpretations of 1+r2(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 + r2 - [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)