For an algebraic number field k and a prime number p (if p = 2, we assume that μ4 ⊂ k), we study the maximal rank ρk of a free pro-p- extension of k. We give various interpretations of 1 + r2(k) - ρk. 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 - ρk is related to the torsion of a certain Iwasawa module, hence to the dualizing module of a certain Galois group (under Leopoldt's conjecture).
@article{urn:eudml:doc:41449,
title = {Sur le rang d'une extension pro-p-libre d'un corps de nombres.},
journal = {Publicacions Matem\`atiques},
volume = {46},
year = {2002},
pages = {201-219},
mrnumber = {MR1904863},
zbl = {1007.11064},
language = {fr},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41449}
}
Lannuzel, Arthur. Sur le rang d'une extension pro-p-libre d'un corps de nombres.. Publicacions Matemàtiques, Tome 46 (2002) pp. 201-219. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41449/